QuestionThe figure shows the electric field inside a cylinder of radius $R=3.3 \mathrm{~mm}$. The field strength is increasing with time as $E=1.0 \times 10^{8} t^{2} \mathrm{~V} / \mathrm{m}$, where $t$ is in s . The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for $t<0$. (Figure 1)
Part A
Part B
Find an expression for the magnetic field strength as a function of time at a distance $r<R$ from the center. Express your answer in teslas as a multiple of product of distance $r$ and time $t$.

Studdy Solution
Use Faraday's law to find the magnetic field strength $B$:
$$E \cdot (2\pi r) = -\frac{d\Phi_B}{dt}$$
Since $\frac{d\Phi_B}{dt} = \frac{d\Phi_E}{dt}$, we have:
$$E \cdot (2\pi r) = -\pi r^2 \cdot 2.0 \times 10^{8} t$$
Solve for $B$:
$$B = \frac{-\pi r^2 \cdot 2.0 \times 10^{8} t}{2\pi r}$$
$$B = -r \cdot 1.0 \times 10^{8} t$$
Express $B$ in teslas:
$$B(t) = -1.0 \times 10^{8} rt \, \text{T}$$
The expression for the magnetic field strength as a function of time is:
$$\boxed{-1.0 \times 10^{8} rt \, \text{T}}$$