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(D) According to the Maxwell-Ampere law,the magnetic field $B$ induced by a time-varying electric field is given by the integral form: $\oint \vec{B}

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proportional to $r$

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(D) According to the Maxwell-Ampere law,the magnetic field $B$ induced by a time-varying electric field is given by the integral form: $\oint \vec{B} \cdot d\vec{l} = \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$.For a cylindrical region of radius $R$ with a uniform electric field $E$ changing with time,we consider an Amperian loop of radius $r$ $(r The electric flux through this loop is $\Phi_E = E \cdot \pi r^2$.Substituting this into the Maxwell-Ampere law: $B(2\pi r) = \mu_0 \epsilon_0 \frac{d}{dt}(E \pi r^2)$.Since $E$ is uniform in space,$\frac{dE}{dt}$ is constant with respect to $r$.$B(2\pi r) = \mu_0 \epsilon_0 \pi r^2 \frac{dE}{dt}$.Solving for $B$,we get $B = \frac{\mu_0 \epsilon_0 r}{2} \frac{dE}{dt}$.Thus,the magnitude of the magnetic field $B$ is proportional to $r$.

$A$ cylindrical region contains a uniform electric field that is along the cylinder axis and is changing with time. If $r$ is the distance from the cylinder axis,the magnitude of the magnetic field within the region is:

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