Lecture 10/5 Chapter 23/24 Flashcards
Self-inductance
o The inductance of a coil depends on
geometric factors
o The SI unit of self-inductance is the
Henry
o 1 H = 1 (V · s) / A
Self-inductance
You can determine an expression for L
L = N(∆ΦB/∆I) = (NΦB/I)
Quick Quiz
A solenoid used in magnetic
resonance imaging (MRI) is 2.4 m
long and 94 cm in diameter, with
1200 turns of superconducting wire.
Find the magnitude of the induced
emf in the solenoid during the 30 s
it takes to “ramp up” the current
from zero to its operating value of
2.3 kA. (µo = 4 π x 10-7 T.m / A)
a. 0.53 V
b. 10.0 V
c. 30.2 V
d. 40.1 V
a. 0.53 V
Inductor in a Circuit
o Inductance can be interpreted
as a measure of opposition to
the rate of change in the current
o Remember resistance R is a
measure of opposition to the
current
o As a circuit is completed, the
current begins to increase, but
the inductor produces an emf
that opposes the increasing
current
o Therefore, the current
doesn’t change from 0 to its
maximum instantaneously
Change of current happens faster than the rate of induction
RL Circuit
o When the current reaches its maximum,
the rate of change and the back emf are
zero
o The time constant, τ, for an RL circuit is
the time required for the current in the
circuit to reach 63.2% of its final value
The time constant depends on R and L
Equation
τ = L/R
The current at any time can be found by
I = (ε /R)(1 - e^(-t/τ)
Energy Stored in Inductors
o The emf induced by an inductor prevents
a battery from establishing an
instantaneous current in a circuit
o The battery has to do work to produce a
current
o This work can be thought of as energy stored
by the inductor in its magnetic field
Energy Stored in Inductors
Equation
UL = ½ L I^2
23-1 Induced Electromotive Force
* Faraday’s experiment:
closing the switch in the primary circuit induces a current in the secondary circuit, but only while the current in the primary circuit is changing.
23-2 Magnetic Flux
Magnetic flux is used in
the calculation of the
induced emf
23-3 Faraday’s Law of Induction:
An emf is induced only when the
magnetic flux through a loop changes with time
- There are many devices that operate on the
basis of Faraday’s law. - An electric guitar pickup
23-4 Lenz’s Law:
– An induced current always flows in a direction
that opposes the change that caused it.
– Therefore, if the magnetic field is increasing,
the magnetic field created by the induced
current will be in the opposite direction;
if decreasing,
it will be in the
same direction.
- This conducting rod completes the circuit. As it
falls, the magnetic flux decreases, and a current
is induced. - The force due to the induced current is upward, slowing the fall.
- Currents can also flow in bulk conductors. These induced currents, called eddy currents, can be powerful brakes.
AC Circuit
o An AC circuit consists of a
combination of circuit
elements and an AC generator
or source
o The output of an AC generator
is sinusoidal and varies with
time according to the following
equation
o Δv = ΔVmax sin 2πƒt
o Δv is the instantaneous
voltage
o ΔVmax is the maximum
voltage of the
generator
o ƒ is the frequency at
which the voltage
changes, in Hz
Resistor in an AC Circuit
o The current and the voltage reach
their maximum values at the same
time
o The current and the voltage are said
to be in phase
o The direction of the current has no effect on the behavior of the resistor
o The rate at which electrical energy is dissipated in the circuit is given by
℘ = = i^(2) R
o where i is the instantaneous current
o the heating effect produced by an AC current with a maximum
value of Imax is not the same as that of a DC current of the same
value
o The maximum current occurs for a small amount of time
rms Current and Voltage
- The voltage and current in an ac circuit both average to zero,
making the average useless in describing their behavior. - We use instead the root mean square (rms); we square the value,
find the mean value, and then take the square root. - 120 volts is the rms value of household ac.
The rms current is the direct current that would dissipate the same
amount of energy in a resistor as is actually dissipated by the AC current
- equation
Alternating voltages can also be discussed in terms of rms values
- equation
Ohm’s Law for a resistor, R, in an AC circuit
o ΔVR,rms = Irms R
o Also applies to the maximum values of v
and i
The average power dissipated in resistor in an
AC circuit carrying a current I is
℘ av = I^2 (rms) R
Quick Quiz
The rms current is equal to the direct current that:
a. produces the same average voltage across a resistor as in an AC circuit.
b. dissipates an equal amount of energy in a resistor at the same rate as in an AC circuit.
c. provides the same average current in a resistor as in an AC circuit.
d. results in the same peak power in a resistor as in an AC circuit.
Capacitors in an AC Circuit
o The impeding effect of a capacitor
on the current in an AC circuit is
called the capacitive reactance
and is given by
Xc = (1/2πfC)
o When ƒ is in Hz and C is in F,
Xc will be in ohms
o Ohm’s Law for a capacitor in an
AC circuit
o ΔVC,rms = Irms Xc
Inductors in an AC Circuit
o The effective resistance of a coil in an AC circuit is called its inductive reactance and is given by
o XL = 2πƒL =ωL
o When ƒ is in Hz and L is in H, XL will be in ohms
o Ohm’s Law for the inductor
o ΔVL,rms = Irms XL
Quick Quiz
The frequency in an AC series circuit is
doubled. By what factor does this change the
capacitive reactance?
a. 1/2
b. 1/4
c. 2
d. 4
The RLC Series Circuit
The current in the circuit is the
same at any time and varies
sinusoidally with time
ΔV, net instantaneous voltage
Phasor Diagrams
- In order to visualize the phase
relationships between the current
and voltage in ac circuits, we
define phasors—rotating vectors
whose length is the maximum
voltage or current, and which rotate
around an origin with the angular
speed of the oscillating current. - The instantaneous value of the
voltage or current represented by
the phasor is its projection on the y
axis.
Phasor Diagram for RLC
Series Circuit
o The voltage across the resistor is on the
+x axis since it is in phase with the
current
o The voltage across the inductor is on the
+y since it leads the current by 90°
o The voltage across the capacitor is on the
–y axis since it lags behind the current by
90°
o The phasors are added as vectors
to account for the phase
differences in the voltages
o ΔVL and ΔVC are on the
same line and so the net
y component is ΔVL - ΔVC
o φ is the phase angle between the current and
the maximum voltage
o The equations also apply to rms values
Quick Quiz
A resistor, inductor, and capacitor are connected in series, each with effective (rms) voltage of 65 V, 140 V, and 80 V, respectively.
What is the value of the effective (rms) voltage of the applied source in the circuit?
a. 48 V
b. 88 V
c. 95 V
d. 285 V
Independence of a Circuit
