Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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2m20001     Induced emf in a coil due to a changing magnetic field     2m20001
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Question

 

A coil of 100 turns and radius 2 cm is in a uniform magnetic field as shown below. The magnetic field changes with time according to the graph.


Figure: A coil in a uniform magnetic field.

  1. What is the emf induced around the coil?
  2. If the coil has a resistance of , what is the current in the coil?
  3. In what direction will current flow in the coil?
  4. What is the direction of the electric field (i.e. the curly electric field) that is induced by the changing magnetic field, at point P (which is located in a +z plane, out in front of the page)?

 



Solution

 

(a) Faraday's Law is:

Since the normal to the plane of the coil is parallel to the magnetic field, and . Thus,

From the graph, you can see that the magnetic field as a function of time is linear. The equation of a straight line is ; therefore, since the intercept is 2 T and the slope is , the magnetic field is . You can check this by substituting and calculating which comes out to zero, as expected from the graph.

Applying Faraday's Law gives:

(b) To calculate the current through the coil, apply Ohm's law. The voltage around the coil is equal to the emf.

(c) Because the emf and the current are positive, this means that from the perspective of the normal to the coil, the current flows counterclockwise around the loop. Or consider Lenz's Law, the induced electric field curls around . The magnetic field at the surface of the coil is in the +x direction. It is decreasing (as shown by the graph), thus is opposite and points to the left, in the -x direction. As a result, is in the +x direction. Point your thumb in the direction of and your fingers curl around the coil, pointing out of the page at the top of the coil and into the page at the bottom of the coil, as shown below.


Figure: Direction of induced current in the coil.

(d) Point P is in a +z plane. The electric field within the coil curls around the magnetic field in the direction given by the right-hand rule. Point your thumb in the direction of and your fingers curl around the coil, pointing out of the page at the top of the coil and into the page at the bottom of the coil, as shown below.


Figure: Direction of the electric field at point P

Note that the curly electric field exists at point P, whether or not their is a coil present. The changing magnetic field induces an electric field regardless of the presence of the coil. Having a coil means that the electric field will exert a force on electrons in the coil, which causes current to flow.