Q7.20P - Introduction to Electrodynamics

Question

Where is $\frac{\mathbf{\partial }\mathbf{B}}{\mathbf{\partial }\mathbf{t}}$  nonzero in Figure 7.21(b)? Exploit the analogy between Faraday's law and Ampere's law to sketch (qualitatively) the electric field.

Step-by-Step Solution

Answer

The term $\frac{\partial \mathrm{B}}{\partial \mathrm{t}}$ is nonzero along the left and right edges of the shaded rectangle.

Step 1: Define Faraday's Law.

Faraday's law is the most fundamental law in Electromagnetic induction, which depicts that whenever a conductor is placed in the magnetic field, which is variable in nature, then an EMF will be induced in the conductor if the conductor has been short circuited, then current will flow in the conductor.

Step 2: Determine the nonzero position of ∂ B ∂ t

The figure concludes that the (inward) flux through the strip on the left is increasing. On the contrary, the (inward) flux passing through the strip on the right decreases, similar to the two current sheets under Ampere's circuital law.

Now, with $B\to E$ and ${\mathrm{\mu }}_0{\mathrm{I}}_{\mathrm{enc}}\to -\frac{\mathrm{d\varphi }}{\mathrm{dt}}$  from equation 7.19, the significance of the negative sign is that the current has been flown out to one on the left, so its field is counter clockwise, and the one on the right is like a current flowing in, so it field is clockwise.

The diagram of the above description is shown below:

Therefore, the term $\frac{\partial \mathrm{B}}{\partial \mathrm{t}}$  is nonzero along the left and right edges of the shaded rectangle.