Question

Figure 30-54 shows two circular regions R1 and R2 with radii r1 = 22.8 cm and r2 = 33.4 cm. In R1 there is a uniform magnetic field of magnitude B1 = 50.8 mT directed into the page, and in R2 there is a uniform magnetic field of magnitude B2 = 76.4 mT directed out of the page (ignore fringing). Both fields are decreasing at the rate of9.40 mT/s. Calculate (in mV) for (a) path 1, (b) path 2, and (c) path 3.

Answer 1

The radius of the first path \(\mathrm{r}_{1}=22.8 \mathrm{~cm}\) the radius of the second path \(\mathrm{r}_{2}=33.4 \mathrm{~cm}\) The magnitude of the magnetic field \(\mathrm{B}_{1}=50.8 \mathrm{mT}\) The magnitude of the magnetic field \(B_{2}=76.4 \mathrm{~m}\) T the change in magnetic fields \(\Delta B=9.40 \mathrm{~m} \mathrm{~T} / \mathrm{s}\) From the given figure

$$ \begin{aligned} \int E_{1} d s &=-\frac{d \phi}{d t} \\ &=-\frac{d(B A)}{d t}=-\left(\pi r^{2}\right) \frac{d B_{1}}{d t} \\ &=-\pi\left(22.8 \times 10^{-2} \mathrm{~m}\right)^{2}\left(9.40 \times 10^{-3} \mathrm{~T} / \mathrm{s}\right) \\ &=-1.53 \times 10^{-3} \mathrm{~V} \text { or }-1.53 \mathrm{~m} \mathrm{~V} \end{aligned} $$

for path 2

$$ \begin{aligned} \int E_{2} d s=&-\frac{d \phi}{d t}=-\frac{d(B A)}{d t}=-\left(\pi r^{2}\right) \frac{d B_{1}}{d t} \\ &=-\pi\left(33.4 \times 10^{-2} \mathrm{~m}\right)^{2}\left(9.40 \times 10^{-3} \mathrm{~T} / \mathrm{s}\right) \\ &=-3.29 \times 10^{-3} \mathrm{~V} \text { or }-3.29 \mathrm{~m} \mathrm{~V} \end{aligned} $$

for path

$$ \int E_{3} d s=\int E_{1} d s-\int E_{2} d s $$

$$ \begin{array}{l} =-(1.53 \mathrm{~m} \mathrm{~V})-(-3.29 \mathrm{mV}) \\ =1.76 \mathrm{~m} \mathrm{~V} \end{array} $$