Question

4. a) A time varying magnetic field B(t) = B0 cos wt pointing out of the page fills the region enclosed by a circle of radius a shown in the figure below. Determine the induced electric field.
b) Do the following fields satisfy all four Maxwell’s equations?
E(t) = E0 sin x sin t
B(t) = B0 cos x cos t

Answer 1

Answer on Question #50908, Physics, Electromagnetism

a) A time varying magnetic field $B(t) = B_{0} \cos \omega t$ pointing out of the page fills the region enclosed by a circle of radius $R$ shown in the figure below. Determine the induced electric field.

Solution

According to Maxwell–Faraday equation $\oint_{L} \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \iint_{S} \vec{B} \cdot d\vec{S}$

$$E 2 \pi R = - \frac {d}{d t} B _ {0} \pi R ^ {2} \cos \omega t \Rightarrow E = \frac {B _ {0} R \omega}{2} \sin \omega t$$

Answer: $E = \frac{B_0R\omega}{2}\sin \omega t$

b) Do the following fields satisfy all four Maxwell’s equations?

$$E (t) = E _ {0} \sin x \cdot \sin t$$ $$B (t) = B _ {0} \cos x \cdot \cos t$$

Solution

According to Gauss's law for magnetism $\nabla \cdot \vec{B} = -\vec{B}_0 \sin x \cos t \neq 0$ for all $x \in \mathbb{R}$.

Answer:

So, the following fields don’t satisfy all four Maxwell’s equations.

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