GRE 9677 problem 2. Faraday's Law of Inductance

GRE Physics practice exam 9677

Problem:
 

2. The circuit shown above is in a uniform magnetic field that is into the page and is decreasing in magnitude at the rate of 150 tesla/second. The ammeter reads:

(A) 0.15 A
(B) 0.35 A
(C) 0.50 A
(D) 0.65 A
(E) 0.80 A

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Solution:

Correct answer: (B) 0.35 A.

Percent of test takers answered correctly: 29%

By Faraday's Law of Inductance, a circuit in the presence of a changing magnetic field ... induces an emf voltage... with the direction of the magnetic field when the magnetic field is decreasing...

Stated in the Maxwell-Farady equation

In the simple arrangement of the problem, the magnetic field is uniform and the line and surface integrals are trivial so the equation has the simple form

With only one loop in the circuit, we reduce all this down to just:

|V| = (|rate of change in magnetic field|) x (area)

= (150 tesla/s) * (0.1 m x 0.1 m) = 0.15 V

Now we use the "right hand rule" to find that the magnetic field is producing a clockwise current. (Right hand thumb points with the magnetic field into the page, fingers curl clockwise with the direction of induced current.)

The voltage generated by the decrease in magnetic field acts against the direction of change...

"if the magnetic field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant." Lenz's Law http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html

So in this case the induced current flows with the magnetic field, in the clockwise direction.
 
The current in the circuit diagram is in a counter-clockwise direction. (The little bar on the DC voltage source is the negative terminal and the big bar opposite is positive. By convention, current flows negative to positive.)

We have all then all the components to the circuit equation:

V_dc - V_induced = I x R

(5.0 V) - (0.15 V) =  I x (10 ohms)

gives the solution (B) for the current

I = 0.35 A

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Sidenotes

Tesla is measure of magnetic field strength = Volt * second / meter^2

Wikipedia (tesla unit) gives some reference numbers for real world magnetic fields:

• 31.869 µT (3.2 × 10⁻⁵ T) – strength of Earth's magnetic field at 0° latitude, 0° longitude
• 5 mT – the strength of a typical refrigerator magnet
• 1.25 T – magnetic flux density at the surface of a neodymium magnet
• 1 T to 2.4 T – coil gap of a typical loudspeaker magnet
• 1.5 T to 3 T – strength of medical magnetic resonance imaging systems in practice, experimentally up to 17 T^[12]

So the 150 Tesla /s decrease in this problem is from a HUGE magnetic field holy crap! Any field with THAT much strength beats anything humans can generate. It would more likely come from some massive collapsing star!

• 45 T – the current (2015) world record for continuous field magnets ^[19]
• 10⁸ - 10¹¹ T (100 MT-100 GT) – magnetic strength of the average magnetar

So this problem seems like an academic textbook-ish exercise, not real world applicable!

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Only 29% of test takers got this question right. I don't think people couldn't think of what to do. I think whether to add or subtract the induced voltage is what threw most test takers off.

A temptation is to point the current opposite to the magnetic field. But Lenz's Law states that the direction of the current opposes the change in magnetic field, not the magnetic field itself! In this case, the induced current went in the same direction as the magnetic field.