A circular region in an xy plane is penetrated by a uniform magnetic field in the positive direction of the z axis.The field’s magnitude B (in teslas) increases with time t (in seconds) according to $$B=at$$, where a is a constant. The magnitude E of the electric field set up by that increase in the magnetic field is given by Figure versus radial distance
r; the vertical axis scale is set by $$E_s=300\mu N/C$$, and the horizontal axis scale is set by $$r_s=4.00cm$$. Find a.
From the "kink" in the graph of Figure , we conclude that the radius of the circular region is $$2.0 \mathrm{cm} .$$ For values of $$r$$ less than that, we have (from the absolute value of Eq. $$\oint \vec{E} \cdot
d\vec{s}=-\dfrac{d\Phi _B}{dt} \text{ (Faraday’s law) }$$)
$$E(2 \pi
r)=\dfrac{d \Phi_{B}}{d t}=\dfrac{d(B A)}{d t}=A \dfrac{d B}{d t}=\pi r^{2} a$$
which
means that $$E / r=a / 2 .$$ This corresponds to the slope of that graph (the
linear portion for small values of $$r$$ ) which we estimate to be 0.015 (in SI
units). Thus,$$a=0.030
\mathrm{T} / \mathrm{s}$$
A circular region in an xy plane is penetrated by a uniform magnetic field in the positive direction of the z axis.The field’s magnitude B (in teslas) increases with time t (in seconds) according to $$B=at$$, where a is a constant. The magnitude E of the electric field set up by that increase in the magnetic field is given by Figure versus radial distance
r; the vertical axis scale is set by $$E_s=300\mu N/C$$, and the horizontal axis scale is set by $$r_s=4.00cm$$. Find a.
Figure a shows a circuit consisting of an ideal battery with emf $$\mathscr{E}=6.00\muV$$, a resistance R, and a small wire loop of area $$5.0cm^2$$. For the time interval t = 10 s to t = 20 s, an external magnetic field is set up throughout the loop.The field is uniform, its direction is into the page in Figure a, and the field magnitude is given by $$B=at$$, where B is in teslas, a is a constant, and t is in seconds. Figure b gives the current $$i$$ in the circuit before, during, and after the external field is set up. The vertical axis scale is set by $$i_s=2.0mA$$. Find the constant a in the equation for the field magnitude.
