Electric Potential and Capacitance

Question 1

Electric Potential Energy

Answer 1

• When a charge moves in an electric field, unless its displacement is always perpendicular to the field, the electric force does work on its charge
  
  • If W_E is the work done by the electric force, then the change in the charge’s electrical potential energy is defined by
    
    • ΔU_E= -W_E (Same as the equation that defined the change in the gravitational potential energy of an object of mass m undergoing a displacement in a gravitational field (ΔU_G= -W_G)

Question 2

Electric Potential

Answer 2

Given a point charge q₁ at a distance r₁ from a fixed charge q₂ moving by some means to a position r₂

• Work done by the electric force during this displacement is given by:
  
  • W_E= -kq₁q₂(1/r₂ - 1/r₁)
    
    • Since ΔU_E= -W_E
      
      • U₂ - U₁= kq₁q₂(1/r₂ - 1/r₁)
      • U= U reference at infinity: U_E= kq₁q₂/r
        
        • If q₁ and q₂ have the same sign, U_E is positive as it takes positive work by an external force to bring these charges together (as they repel) In contrast if q₁ and q₂ have opposite sign, U_E is negative
• Everything naturaly tends toward lower potential energy
  
  • Positive charges tend toward lower potential and negative charges tend toward higher potential

Question 3

Voltage

Answer 3

The change in electric potential, ΔV is defined as this ratio: ΔV= ΔU_E/q

• Electric potential is electrical potential energy per unit charge
  
  • Units of Joules per coulomb
• One joule per coulomb is called one volt (abbreviated V)
  
  • 1 J/C= 1 V
• In an electric field created by a point source charge Q, the electric potential at a distance r from Q is
  
  • V= kQ/r
    
    • Potential depends on the source charge making the field and the distance from it

Question 4

Equipotential Surfaces

Answer 4

• Like potential energy, potential is scalar
  
  • Direction doesn’t matter, at any point on a sphere for a specific r from Q, the potential is constant
    
    • These spheres around Q are called equipotential surfaces, and they’re surfaces of constant potential
      
      • Their cross sections in any plane are circles and are, therefore, perpendicular to the electric field lines (the equipotentials are always perpendicular to the electric field lines)

Question 5

Capacitors

Answer 5

Two conductors separated by some distance, carry equal but opposite charges, +Q and - Q; this pair of conductors make up a system called a capacitor

• Work must be done to create this separation of charge, and as a result, potential energy is stored
• Capacitors are basically storage devices for electrical potential energy
• Most common conductors are parallel-plate capacitors which are parallel metal plates or sheets
  
  • Assuming distance d between the plates is small compared to the dimensions of the plates, in which chase the electric field between the plates is uniform
    *

Question 6

Capacitance

Answer 6

The ratio of Q to the voltage, ΔV, for any capacitor is called its capacitance (C)

C= Q/ΔV

• Capacitance is a measure of the capacity for holding charge
  
  • The greater the capacitance, the more charge can be stored on the plates at a given potential difference
• The capacitance of any capcitor depends only on the size, shape, and separation of conductors
• The units of C are coulombs per volt; one coulomb per volt is renamed one farad (abbreviated F)
  
  • 1 C/V= 1 F

Question 7

Combination of Capacitors

Answer 7

• When a capcitor charges up, work must be done by an external force (E.G.) a battery
  
  • This increases the potential energy stored by the capcitor
    
    • The potential energy stored is given by:
      
      • PE= 1/2QΔV= 1/2CV²= 1/2Q²/C
• Capacitors are arranged in combination in electric circuits of the types
  
  • Parallel combination
  • Series Combination

Question 8

Parallel Combination

Answer 8

A collection of capacitors are said to be in parallel if they all share the same potential difference

• The top plates are connected by a wire and form a single equipotential; the same is true for the bottom plates
  
  • The potential difference across one capacitor is the same as the potential difference across the other capacitor
• To find the capacitance of a single capacitor that would perform the same function as this combination, and if the capacitances are C₁ and C₂, then the charge on the first capacitor is Q₁= C₁ΔV and the charge on the second capacitor is Q₂= C₂ΔV
  
  • The total charge on the combination is Q₁ + Q₂, the equivalent capacitance C_p must be:
    
    • C_p= Q/ΔV= (Q₁ + Q₂)/ΔV= Q₁/ΔV + Q₂/ΔV or
    • C_p= C₁ + C₂
      
      • Can be extended to more than 2 capacitors
        
        • The equivalent capacitance of a collection of capacitors in parallel is found by adding the individual capacitances

Question 9

Series Combination

Answer 9

A collection of capacitors are said to be in series if they all share the same charge magnitude

• When a potential difference is applied, negative charge will be deposited on the bottom plate of the bottom capacitor; this will push an equal amount of negative charge away from the top plate of the bottom capacitor toward the bottom plate of the top capacitor
  
  • When the system has reached equilibrium, the charges on all the plates will have the same magnitude
• If the top and bottom capacitors have capacitances of C₁ and C₂ respectively then the potential difference across the top capacitor is ΔV₁= Q/C₁ and the potential difference across the bottom capacitor is ΔV₂= Q/C₂
  
  • The total potential difference across the combination is ΔV₁ + ΔV₂ which equals ΔV therefore the equivalent capacitance, C_s is given by
    
    • C_s= Q/ΔV= Q/(ΔV₁ + ΔV₂)= Q/(Q/C₁ + Q/C₂)= 1/(1/C₁ + 1/C₂)
      
      • Rewritten as: 1/C_s= 1/C₁ + 1/C₂
      • Can be extended to more than 2 capacitors
      • The reciprocal of the capacitance of a collection of capacitors in a series is found by adding the reciprocals of the individual capacitances

Question 10

Dielectrics

Answer 10

One method of keeping the plates of a capacitor apart, which is necessary to maintain charge separation and store potential energy, is to insert an insulator (called dialectric) between the plates

• A dielectric always increases the capacitance of a capacitor
• Although the dielectric is not a conductor, the electric field that existed between the plates causes the molecules within the dielectric material to polarize; there is more electron density on the side of the molecule near the positive plate
  
  • A layer of negative charge is formed along the top surface of the dielectric and a layer of positive charge along the bottom surface
    
    • This separation of charge induces its own electric field (E_i) within the dielectric, which opposes the original electric field, E, within the capacitor
• E_{with dielectric}= E_{without dielectric} - E_i= E/ϰ
  
  • ϰ is a factor by which electric field is reduced
• C_{with dielectric}= ϰC_{without dielectric}
  
  • The value of ϰ, called the dielectric constant, varies from material to material, but is always greater than 1