Faradays Law Examples
We will now solve a number of examples that involve Faraday’s law.
Example N3.1: Determine the emf around the contour c that includes the moving slider in the figure below.
Apply Faraday’s law to the contour c in the direction shown:
ym changes with time because the surface s increases with t as the slider moves to the right. The magnetic flux density ` , however, is not changing with time. Therefore,
Note that we have ignored `ind created by current induced in the wire. This would be a reasonable assumption for a high resistance wire, for example. Another way to approach this problem is with the Lorentz force equation In this problem, `e=0 while
Both approaches give the same result. Physically, this emf will cause charges to move in the wire since ` / q = v ´ ` (like an electric field!):
The equivalent circuit model for this is:
Note in the construction of this equivalent circuit that:
1. positive current is in the direction of positive emf,
2. current enters the negative terminal of the source.
Example N3.2: Determine the voltage measured by the highimpedance
voltmeter in the circuit below.
For the direction of c shown,
This emf serves as a voltage source in the equivalent lumpedelement circuit:
It is very important to note the polarity of the equivalent emf voltage source in this circuit. The contour c was initially chosen clockwise. Consequently, the current I must enter the ‘-’ terminal of this voltage source, as shown. Therefore,
Summary of steps for the solution:
1. Begin with the actual physical circuit with dimensions.
2. Pick a direction for the contour c (used in the emf equation).
3. Compute emf (- dy/ dt ).
4. Construct the lumped element circuit (no dimensions) inserting the appropriate equivalent emf source(s).
5. Solve the lumped element circuit for the desired voltage(s) and current(s).
Example N3.3: Determine the voltage measured in the circuit of the previous example, but with the high-impedance voltmeter and leads oriented as shown below.
Equivalent lumped element circuit:
Using KVL around loop L gives:
Something very strange has just happened. The measured voltages in these last two example problems have very different values 8 /5 V and 12/ 5 V)!
Voltage May Not Be Unique
This is an example illustrating that “voltage” is not a unique quantity for time-varying electromagnetic fields. Measured voltages may depend on how the leads of the voltmeter (or oscilloscope) are laid out and the time rate of change of the magnetic flux through this measuring loop.
To be more specific, why was the measured voltage different in the previous two examples? Because electric scalar potential is not unique for time-varying fields. Consider Faraday’s law
and the contours shown below:
Along c, (1) is
This is generally not zero unless:
. no time variation, or
.no magnetic flux linkage through s.
Therefore, we conclude that is not conservative. Since it is not conservative, we cannot (uniquely!) define a scalar potential as E e as was done in Ch. 3. But, if we did define
we can see from (2) that Ñje depends on the contour taken, and consequently Ñje is not unique! You will generally compute (or measure) a different voltage depending on the path used for the integration (or how the measurement leads are laid out)! At “high” frequencies, this non-uniqueness of scalar electric potential can adversely affect oscilloscope measurements. A typical oscilloscope is single-ended input meaning that the black alligator clip on the scope probe is earth ground.
Measurements taken without connecting the scope probe alligator clip to ground will be correct at “low” frequencies (remember the scope is single-ended input).
Without connecting the alligator clip, the “emf loop” can be quite large since the ground loop passes through the o'scope power cord, through the lab bench electrical power wiring, back through the function generator power cord, through the function generator test leads then back to the circuit.
At high frequencies (say roughly > 100 MHz), this emf loop can cause the measured voltage to change as the leads are moved around! Not desirable. With the alligator clip attached, the “emf loop” is greatly reduced in size