(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

An electric field is directed out of the page within a circular region of radius R = 3.00 cm. The field magnitude is [itex]E = (0.500 V/ms)(1 - \frac{r}{R})t[/itex], where t is in seconds and r is the radial distance (r≤R). What is the magnitude of the induced magnetic field at a radial distance of 2.00 cm?

2. Relevant equations

Maxwell - Ampere's Law

[itex]\oint \vec{B} \cdot d\vec{s} = μ_0ε_0 \frac{d\Phi_B}{dt}[/itex]

3. The attempt at a solution

Since B and ds are parallel, their dot product will equal Bds, or since ds is a circle at radius r, [itex]2\pi rB[/itex]. I got caught up on the left-hand side of the equation. I know I can simplify it to:

[itex]μ_0ε_0 \frac{d}{dt}\int \vec{E} \cdot d\vec{A}[/itex], which further simplifies to (since E and dA are parallel):

[itex]μ_0ε_0 \frac{d}{dt}\int E dA[/itex]

I'm lost as to how to simplify that integral. I think I could do a double integral to solve to surface integral, however, this is an introductory physics course only requiring Calculus 1 and 2, so I wouldn't have assumed such integrals would be necessary.

Also, assuming I've simplified correctly, would I integrate, then take the time-derivative, or could I take the time derivative of E first, then integrate?

Thanks!

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# Induced Magnetic Field in a Non-Uniform Electric Field

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