Electricity (2) Flashcards
Capacitors
Capacitors:
- Store energy in circuits by storing electric charge, creating electric potential energy.
- Consist of two conductive plates separated by a dielectric, preventing charge flow.
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Parallel Plate Capacitor:
- Q: Charge stored on the plates.
- V: Potential difference across the plates.
- One plate holds +Q, the other –Q, with a potential difference V between them.
Capacitance
Capacitance:
- Defined as the charge stored per unit potential difference.
- Units: Farads (F), often in smaller units like μF, nF, or pF.
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Capacitance Equation:
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C = Q / V where:
- C = capacitance.
- Q = charge stored.
- V = potential difference.
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Key Points:
- Charge stored refers to the magnitude of charge on each plate or surface of a spherical conductor.
- Higher capacitance means the capacitor can store more charge for the same potential difference.
Use of capacitors
- Energy storage: Capacitors store electric potential energy for various applications.
- Camera flashes: Provide a bright flash of light during discharge.
- Smoothing currents: Stabilize current in electronic circuits.
- Backup power: Supply power during unexpected outages for memory devices like calculators.
- Timing circuits: Used in electronic timers for precise operations.
Charging capacitor
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Initial Setup:
- A circuit with a battery (e.m.f. ε), resistor (R), capacitor (C), and switch connected in series.
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Switch Closed:
- Electrons flow from the negative terminal of the battery, through the resistor, to the negative plate of the capacitor.
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Plate Charging:
- The positive terminal pulls electrons from one plate, leaving it positively charged.
- The negative terminal pushes electrons onto the other plate, making it negatively charged.
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Insulator Role:
- The insulator between the plates prevents charge flow, forcing charge to accumulate.
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Electrostatic Repulsion:
- As negative charge builds, it repels incoming electrons, slowing the flow of charge.
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Current and Voltage:
- Current decreases exponentially over time.
- Potential difference (V) across the plates increases as charge accumulates, eventually matching the supply voltage.
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Fully Charged:
- The capacitor stops charging when it reaches maximum charge, determined by its capacitance (C) and the supply voltage.
Discharging Capacitors
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Initial Setup:
- A circuit with a resistor (R), switch, and capacitor (C) in series. No power supply is present.
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Switch Closed:
- The potential difference (V) across the capacitor causes a current (I) to flow through the circuit.
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Current Flow:
- Electrons flow from the negative plate of the capacitor, through the resistor, to the positive plate, reducing charge on both plates.
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Exponential Decay:
- Current, potential difference, and charge decrease exponentially over time.
- The rate of decrease is proportional to the amount remaining.
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Discharge Completion:
- The capacitor is fully discharged when potential difference (V) and current (I) fall to zero.
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Energy Dissipation:
- The electrical energy stored in the capacitor is transferred to thermal energy in the resistor.
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Graphs:
- Current, potential difference, and charge follow an identical exponential decay pattern over time.
Energy Stored by a Capacitor
Charge and Potential Difference:
- Charge (Q) is directly proportional to the potential difference (V), forming a straight-line graph.
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Energy Stored:
- The electrical energy stored in the capacitor is represented by the area under the potential-charge graph, forming a triangle.
Capacitors in Series
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Charge and Potential Difference:
- The potential difference (p.d.) is shared between capacitors, but each stores the same charge (Q).
- A negative charge on the left plate of C₁ induces an equal positive charge on its right plate.
- This causes a negative charge on the left plate of C₂, equal to the positive charge on its right plate.
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Total Potential Difference:
- Vtotal = V1 + V2
- Substituting V = Q/C
Vtotal = (Q/C1) + (Q/C2)
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Total Capacitance:
- Since current (and charge Q) is the same in series, Q cancels out:
1/Ctotal = (1/C1) + (1/C2)
- Since current (and charge Q) is the same in series, Q cancels out:
Capacitors in Parallel
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Potential Difference:
- All capacitors have the same potential difference (p.d.) across them.
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Charge Distribution:
- The current splits across each junction, so the charge stored on each capacitor is different.
- The total charge (Q) is the sum of the charges on each capacitor:
Q = Q1 + Q2.
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Charge on Each Capacitor:
Q1 = C1V and Q2 = C2 V, where V is the common p.d.- Therefore, Qtotal = (C1 + C2) V
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Total Capacitance:
- Ctotal = C1 + C2 + C3 + ….
Time Constant
Time to Half (t1/2):
- The time it takes for the charge, current, or voltage of a discharging capacitor to decrease to half its initial value.
- Equation: t1/2 = ln(2) τ
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Time Constant (τ):
- Measures how long it takes for the charge, current, or voltage of a discharging capacitor to decrease to 37% of its original value, or for a charging capacitor to rise to 63% of its maximum value.
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Equation: τ = RC where:
- R = resistance (Ω).
- C = capacitance (F).
How to verify if potential difference (V) or charge (Q) on a capacitor decreases exponentially?
Constant ratio method:
- Plot a V-T graph and check if the time constant is constant,
- when t = τ the potential difference on the capacitor will have decreased to approximately 37% of its original value
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Logarithmic graph method
- Plot a graph of ln(V )against time (t) and check if a straight-line graph is obtained.
Charging and Discharging graphs
Charging Graphs:
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p.d. and Charge vs. Time:
- Both graphs have identical shapes, starting at 0 and increasing to a maximum value.
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Current vs. Time:
- An exponential decay curve, starting at 0 and decreasing exponentially.
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Discharge Graphs:
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Current, p.d., and Charge vs. Time:
- All graphs show exponential decay curves with decreasing gradients.
- Initial values (I₀, V₀, Q₀) start on the y-axis and decrease exponentially.
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Rate of Discharge:
- High resistance: Slower discharge, current decreases, capacitor discharges slowly.
- Low resistance: Faster discharge, current increases, capacitor discharges quickly.
Electric field
A region where a unit charge experiences an electrostatic force.
Electric Field Lines In a uniform electric field
Uniform Electric Field:
- Field lines are equally spaced at all points.
- Electric field strength is constant at all points.
- The force on a test charge has the same magnitude and direction everywhere.
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Electric Field Between Parallel Plates:
- When a potential difference is applied, the plates become charged.
- The electric field between the plates is uniform.
- The field beyond the edges is non-uniform.
- Field lines are directed from the positive to the negative plate.
- A uniform electric field has equally spaced field lines.
Electric Field Lines In a radial electric field:
Radial Electric Field:
- Field lines are equally spaced near the charge but spread out with distance.
- Electric field strength and force on a test charge decrease with distance from the charge.
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Around a Point Charge:
- The field is radial, with lines:
- Radially inwards for negative charges.
- Radially outwards for positive charges.
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Field Strength:
- The field is stronger where lines are closer together and weaker where lines are further apart.
Electric Field Strength
- Electrostatic force per unit positive charge
- acting on a charge at a specific point
- or on a stationary point charge
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- Describes how strong or weak an electric field is
Coulomb’s Law
Electric Fields and Coulomb’s Law:
- All charged particles produce an electric field, exerting a force on other charges within range.
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Coulomb’s Law:
- The force between two charges is:
- Proportional to the product of their charges (Qq).
- Inversely proportional to the square of their separation (r2).
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Charge Interactions:
- Like charges (Qq > 0) Repulsion.
- Opposite charges (Qq < 0) Attraction.
Electric Field strength of a Point Charge
Radial Field
- Charged sphere acts as a point charge
- Follows an inverse square law (1/r²)
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Direction
- Towards a negative charge
- Away from a positive charge.
Electric Field Strength in a Uniform Field
Electric Field Strength in a Uniform Field:
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Equation: E = V/d
where:- E = electric field strength.
- V = potential difference.
- d = distance between plates.
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Key Points:
- Greater voltage (V): Results in a stronger field (E).
- Greater separation (d): Results in a weaker field (E).
- Does not apply to a radial field around a point charge.
- Field direction: From the positive to the negative plate.
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Derivation:
- Work done on a charge ( Q ):
W = ΔV × Q - Work done as force × distance:
W = F × d - Equating the two:
F × d = ΔV × Q. - Rearranging:
F/Q = ΔV/d. - Since E = F/Q
E = ΔV/d.
Relative permittivity
Permittivity:
- Measures how easy it is to generate an electric field in a material.
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Relative permittivity (εr) (dielectric constant)
- ratio of permittivity of a material to permittivity of free space (ε0):
- εr = ε/ε0.
- Dimensionless because it’s a ratio of two quantities with the same unit.
Effect of Dielectric on Capacitance
Dielectric in a Capacitor:
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Polar Molecules:
- Align with the applied electric field, creating an opposing electric field.
- This reduces the overall electric field, lowering the potential difference between the plates.
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Permittivity:
- Reflects how well polar molecules align with the field; higher alignment = higher permittivity.
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Parallel Plate Capacitor:
- Plates of area A, separated by distance d, with a dielectric of permittivity ε between them.
- The reduction in potential difference increases the capacitance of the plates.
Motion of Charged Particles in an Electric Field
Charged Particles in an Electric Field:
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Force and Motion:
- Charged particles experience a force, causing them to move.
- In a uniform electric field, particles move parallel to the field lines (direction depends on charge).
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Perpendicular Motion:
- A particle moving perpendicular to the field follows a parabolic trajectory due to the constant force.
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Deflection:
- Positive charges: Deflect towards the negative plate.
- Negative charges: Deflect towards the positive plate.
- Deflection depends on mass, charge, and speed:
- Heavier particles deflect less.
- Larger charges and slower particles deflect more.
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Formulas:
- Force: F = EQ
- Work Done: W = Fd
- Kinetic Energy: The force increases the particle’s kinetic energy, causing constant acceleration (Newton’s second law).
- Perpendicular Velocity: If the particle’s velocity has a component perpendicular to the field, it remains unchanged (Newton’s first law).
Electric Potential
Electric Potential:
- Defined as the work done per unit positive charge to bring a test charge from infinity to a defined point.
- A scalar quantity (no direction) but can be positive, negative, or zero:
- Positive around an isolated positive charge.
- Negative around an isolated negative charge.
- Zero at infinity.
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Total Electric Potential:
- The total potential at a point from multiple charges is the sum of the potentials from each individual charge.
The graph of potential V against distance r for a negative or positive charge
Electric Potential for a Positive Charge:
- As distance (r) decreases, electric potential (V) increases.
- More work is required to overcome the repulsive force as the test charge moves closer.
- V starts positive and increases as r decreases, approaching zero as r approaches infinity.
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Electric Potential for a Negative Charge:
- As distance (r) decreases, electric potential (V) decreases (becomes more negative).
- Less work is required due to the attractive force, which pulls the test charge closer.
- V starts negative and decreases (in magnitude) as r decreases, approaching zero as r increases towards infinity.
Electric Potential Energy
Work in an Uniform Electric Field:
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Work is done when a charge moves through an electric field.
- Positive charge: Work is done when it moves against the field.
- Negative charge: Work is done when it moves with the field.
Key Points
- The work done equals the change in electric potential energy.
- When the electric potential is zero, the electric potential energy is also zero.
- The work done depends on the distance the charge moves in the field.
- q = charge being moved; Q = charge producing the potential. Do not confuse the two in calculations.
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Work in a Radial Field:
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Work is required to:
- Move a positive charge closer to another positive charge (overcoming repulsion).
- Move a positive charge away from a negative charge (overcoming attraction).
Key points
- Potential energy increases when moving a charge towards a repelling charge and decreases when moving away from an attracting charge.
Force-Distance Graph for a Point Charge
Force-Distance Graph:
- Force (F) values are all positive.
- As r increases, F follows a 1/r² relation (inverse square law).
- The area under the graph represents work done (ΔW).
- The graph shows a steep decline as r increases.
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Estimating Area:
- Use methods like counting squares or summing areas of trapeziums.
Electric field between two point charges
Opposite Charges:
- Field lines are directed from the positive charge to the negative charge.
- As the charges get closer, the attractive force becomes stronger.
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Same Type Charges:
- Field lines are directed away from two positive charges or towards two negative charges.
- As the charges get closer, the repulsive force becomes stronger.
- A neutral point exists at the midpoint where the resultant electric force is zero.
Capacitance of an Isolated Sphere
Capacitance of a Charged Sphere:
- Defined as the charge per unit potential at the surface:
C = Q / V
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Key Equations:
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Potential of an Isolated Point Charge:
V = Q / (4πε₀R) -
Capacitance of an Isolated Sphere:
C = 4πε₀R
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Variables:
- Q = charge on the sphere (considered as a point charge at its center).
- R = radius of the sphere.
- ε₀ = permittivity of free space.
Electric Fields vs Gravitational Fields
Similarities Between Gravitational and Electrostatic Forces:
- Both follow the inverse square law.
- Field lines around a point mass and a negative point charge are identical.
- Field lines in uniform gravitational and electric fields are identical.
- Field strengths in a radial field have a 1/r relationship.
- Potential in both fields has a 1/r relationship.
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Equipotential surfaces are:
- Spherical around a point mass or charge.
- Parallel in uniform fields.
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Work done is the product of:
- Mass and change in gravitational potential.
- Charge and change in electric potential.
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Differences:
- Gravitational force acts on mass; electrostatic force acts on charge.
- Gravitational force is always attractive; electrostatic force can be attractive or repulsive.
- Gravitational potential is always negative; electric potential can be negative or positive.
Magnetic Fields
Magnetic Field:
- A field of force created by moving electric charges or permanent magnets, also called a B-field.
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Sources of Magnetic Fields:
- Permanent magnets produce magnetic fields.
- Current-carrying wires create magnetic fields due to the movement of electrons (stationary charges do not produce magnetic fields).
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Observing Magnetic Fields:
- Although invisible, their effects can be observed through:
- The force acting on magnetic materials (e.g., iron).
- The movement of a needle in a plotting compass.
Field Lines in a Current-Carrying Wire
Magnetic Field Around a Current-Carrying Wire:
- Field lines are circular rings centered on the wire.
- The field is strongest near the wire and weakens with distance.
- Reversing the current reverses the direction of the field lines.
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Maxwell’s Right-Hand Screw Rule:
- Point your thumb in the direction of the conventional current (positive to negative).
- The curled fingers indicate the direction of the magnetic field around the wire.
Magnetic Field Lines in Solenoids and Coils
Field Lines in a Solenoid:
- Electromagnets use solenoids (coils of wire) to concentrate magnetic fields.
- One end becomes the north pole, and the other becomes the south pole.
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Magnetic field lines resemble those of a bar magnet:
- Emerge from the north pole.
- Return to the south pole.
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Field Lines in a Flat Circular Coil:
- Behaves like a single loop of a solenoid.
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Field lines:
- Emerge from one side (north pole).
- Return to the other side (south pole).
- Multiple coils in a solenoid combine to create a stronger, uniform field.
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Right-hand thumb rule:
- Thumb shows the magnetic field direction.
- Fingers show the current direction.
Factors Affecting the Magnetic Field Strength
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Add a Ferrous Core:
- Use a core made from a ferrous (iron-rich) material (e.g., an iron rod).
- When current flows, the core becomes magnetised, increasing the field strength by several hundred times.
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Add More Turns to the Coil:
- Concentrates the magnetic field lines, increasing the field strength.
Fleming’s Left-Hand Rule
Fleming’s Left-Hand Rule:
- Determines the direction of magnetic force on a moving charged particle in a magnetic field:
- First Finger: Direction of the magnetic field.
- Second Finger: Direction of conventional current (velocity of a moving positive charge).
- Thumb: Direction of the magnetic force.
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Magnetic Field Direction in 3D:
- Dots (tip of an arrow): Magnetic field coming out of the page.
- Crosses (back of an arrow): Magnetic field going into the page.
Magnetic Flux Density
- Defined as the force acting per unit current per unit length on a current-carrying conductor placed perpendicular to the magnetic field.
- Units: Tesla (T), where 1 T = 1 N m⁻¹ force on a conductor carrying 1 A current normal to the field.
- Also referred to as magnetic field strength.
Force on a Current-Carrying Conductor
Magnetic Force on a Current-Carrying Conductor:
- A current-carrying conductor produces its own magnetic field.
- An external magnetic field exerts a magnetic force on the conductor.
- The maximum force occurs when the current is perpendicular to the magnetic flux lines.
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Magnitude of Magnetic Force (F):
- Proportional to:
- Current (I).
- Magnetic flux density (B).
- Length of conductor in the field (L).
- Sine of the angle (θ) between the conductor and the magnetic flux lines.
- No force is experienced if the current is parallel to the magnetic field.
Force on a Moving Charge
Magnetic Force on a Moving Charged Particle:
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Equation: F = BQv, where:
- F = magnetic force (N).
- B = magnetic flux density (T).
- Q = charge of the particle (C).
- v = speed of the particle (m/s).
- This is the maximum force when F, B, and v are mutually perpendicular.
- If the particle travels parallel to the magnetic field, it does not experience a magnetic force.
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Force at an Angle θ:
- When the particle moves at an angle θ to the magnetic field lines:
F = BQv sin θ
Motion of a Charged Particle in a Magnetic Field
Circular Motion of a Charged Particle in a Magnetic Field:
- A charged particle in a uniform magnetic field perpendicular to its motion travels in a circular path because:
- The magnetic force (F) is always perpendicular to its velocity (v), causing circular motion.
- The magnetic force always points towards the center of the circular path.
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Centripetal Force:
- Provides the force required for circular motion:
mv² / r = BQv - Rearranging for the radius r:
r = mv / BQ
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Key Points:
- Faster particles (v): Move in larger circles r ∝ v
- Greater mass (m): Move in larger circles r ∝ m
- Greater charge (q): Move in smaller circles r ∝ 1 / q
- ## Stronger magnetic field (B): Move in smaller circles r ∝ 1 / B
- Calculated using Newton’s second law:
F = ma
Charged Particles in a Velocity Selector
Velocity Selector:
- Filters charged particles by using perpendicular electric and magnetic fields to allow only particles with a specific velocity to pass through.
- Used in devices like mass spectrometers to create a beam of particles moving at the same speed.
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Setup:
- Two oppositely charged plates create an electric field (E).
- A magnetic field (B) is applied perpendicular to the electric field.
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Force Balance:
- Electric force: FE = EQ (independent of velocity).
- Magnetic force: FB = BQv (depends on velocity).
- For a particle to pass through undeflected, the forces must balance:
FE = FB. - The selected velocity is:
v = E / B.
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Deflection:
- Particles with velocities different from v are deflected and removed from the beam.
Magnetic Flux
Magnetic Flux (Φ):
- Defined as the product of magnetic flux density (B) and the cross-sectional area (A) perpendicular to the magnetic field:
Φ = B A - Units: Webers (Wb).
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Key Points:
- Maximum flux: Occurs when the magnetic field lines are perpendicular to the area.
- Minimum flux: Occurs when the magnetic field lines are parallel to the area.
- Represents the amount of magnetic field passing through a given area.
Magnetic Flux Linkage
Magnetic Flux Linkage:
- Commonly used for solenoids with N turns of wire.
- Defined as the product of magnetic flux (Φ) and the number of turns (N) in the coil:
ΦN = Φ × N = B × A × N** where:- B = magnetic flux density.
- A = cross-sectional area of the coil.
- N = number of turns in the coil.
- Units: Weber turns (Wb turns).
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General Equation:
-ΦN = B × A × N × cos(θ) where θ is the angle between the magnetic field lines and the normal to the coil.
Faraday’s Law
- The magnitude of the induced e.m.f. (electromotive force) is directly proportional to the rate of change of magnetic flux linkage.
- The equation form of Faraday’s Law is:
ε = Δ(Nɸ) / Δt
Where:- ε = induced e.m.f (V)
- Δ(Nɸ) = change in flux linkage (Wb turns)
- Δt = time interval (s)
Lenz’s Law
Lenz’s Law:
- The induced e.m.f. produces effects that oppose the change causing it.
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Example:
- If a north pole approaches the coil, the induced e.m.f. creates an opposing north pole to repel the incoming magnet.
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Right-Hand Grip Rule:
- Curl fingers around the coil in the direction of the current.
- The thumb points along the direction of the magnetic flux (north to south).
- In this case, the current flows anti-clockwise, inducing a north pole to oppose the incoming magnet.
Induced E.m.f.
Faraday’s Law with Lenz’s Law:
The equation that combines Faraday’s Law and Lenz’s Law is written as:
ε = - Δ(Nɸ) / Δt
- The negative sign represents Lenz’s Law.
EMF Inducted in a Rotating Coil
Induced E.M.F. in a Rotating Coil:
- When a coil rotates in a uniform magnetic field, the magnetic flux through the coil changes, causing the induced e.m.f. to change.
- The e.m.f. is:
- Maximum when the coil cuts through the most magnetic field lines (normal to the coil is perpendicular to the field).
- Zero when the coil is aligned with the field (normal to the coil is parallel to the field).
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E.M.F. Formula:
- ε = BANω sin(θ)
- where θ = ωt
- The e.m.f. varies sinusoidally and is 90° out of phase with the flux linkage.
Alternators
Alternator:
- A device that converts mechanical energy into electrical energy using a rotating coil in a magnetic field.
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Components:
- Coil: A rectangular coil rotates within a uniform magnetic field, generating electricity.
- Slip Rings: Metal rings that rotate with the coil, maintaining continuous electrical contact.
- Brushes: Metal brushes press against the slip rings to transfer current to the external circuit.
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Operation:
- As the coil rotates, it cuts through magnetic field lines, inducing a potential difference (voltage) across the coil.
- The induced current changes direction as the coil spins, creating an alternating current (AC).
- The meter pointer deflects first in one direction, then the opposite, as the current reverses.
- The induced voltage and current alternate direction continuously, producing a steady AC waveform.
Dynamos
Dynamo:
- A direct-current (D.C.) generator that uses a split-ring commutator instead of slip rings.
- Consists of a rotating coil in a magnetic field.
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Operation:
- The split-ring commutator ensures the current stays in one direction.
- The induced potential difference varies only in the positive region of the graph, never reversing direction.
- The current is always positive (or negative), never alternating.
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Key Process:
- As the coil rotates, it cuts through magnetic field lines, inducing a potential difference between the coil’s ends.
- The split-ring commutator changes the coil’s connection to the brushes every half turn, keeping the current in the same direction.
- This change occurs when the coil is perpendicular to the magnetic field lines.
Transformer Basics
Transformer:
- Changes high alternating voltage at low current to low alternating voltage at high current, and vice versa.
- Increases transmission efficiency by reducing heat energy loss in power lines.
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Power Loss:
- Given by P = I²R, so reducing current reduces power loss during transmission.
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Applications:
- Used in the National Grid for efficient power transmission.
- Step-up transformers: Increase voltage and decrease current for long-distance transmission.
- Step-down transformers: Reduce voltage and increase current for local use near homes and businesses.
Transformer Components and Functioning
Step-Up Transformer:
- The secondary coil has more turns than the primary coil, increasing voltage.
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Operation:
- The primary coil is powered by an alternating current (AC), creating a changing magnetic field in the iron core.
- The changing magnetic field induces an e.m.f. in the secondary coil.
- The secondary voltage depends on the number of turns:
- More turns = step-up (voltage increases).
- Fewer turns = step-down (voltage decreases).
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Components:
- Primary coil, secondary coil, and soft iron core.
- The soft iron core focuses and directs the magnetic field between the coils, and is used because it can be easily magnetised and demagnetised.
Eddy Currents
Transformer Efficiency:
- Transformers are not 100% efficient, and power loss occurs due to various factors.
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Eddy Currents:
- Looping currents in the core caused by the changing magnetic flux.
- Effects:
- Generate heat, leading to energy loss.
- Create a magnetic field that opposes the inducing field, reducing field strength.
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Reducing Eddy Currents:
- The core is laminated, with layers separated by thin insulating material to prevent current flow.