Question

In Figure 22.72 a toroid has a rectangular cross section with an inner radius ${\mathbf{r}}_{\mathbf{1}}\mathbf{=}\mathbf{9}\mathbf{cm}$, an outer radius ${\mathbf{r}}_{\mathbf{2}}\mathbf{=}\mathbf{12}\mathbf{cm}$, and a height $\mathbf{h}\mathbf{=}\mathbf{5}\mathbf{cm}$, and it is wrapped around by many densely packed turns of current-carrying wire (not shown in the diagram). The direction of the magnetic field inside the windings is shown on the diagram. There is essentially no magnetic field outside the windings. A wire is connected to a sensitive ammeter as shown.

The resistance of the wire and ammeter is $\mathbf{R}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{cm}$.

The current in the windings of the toroid is varied so that the magnetic field inside the windings, averaged over the cross section, varies with time as shown in Figure 22.73:

Make a careful graph of the ammeter reading, including sign, as a function of time. Label your graph, and explain the numerical aspects of the graph, including signs.

Short answer

The graph for the ammeter reading is shown in Figure below:

Step-by-step solution

Write the given data

Refer to the diagram in 22.72 and 22.73 in the text book.

Determine the formula:

Write the formula for the emf induced in the wire that passes through the toroid of the inner radius $r_{1}and{r}_2$with the height is given as:

$\mathrm{e}=-\mathrm{A}\frac{\mathrm{dB}}{\mathrm{dt}}$ ……. (1)

Determine the magnitude of the voltage as the function of the time.

Determine the area of the toroid as:

$A=2\pi \left(r_2-r_1\right)h$

Substitute the values and solve as:

$$\begin{gathered} A=2\left(3.14\right)\left(12\text{cm}-9\text{cm}\right)\left(\text{5}\text{cm}\right) \\ =94.2{\text{cm}}^2 \end{gathered}$$

Consider the current in the circuit is of the form as:

$I=\frac{e}{R}$

Resolve the equation (1) as follows:

$I=\frac{-A}{R}\frac{dB}{dt}$

Substitute the values and solve as:

$$\begin{gathered} I=-\frac{9.42\times {10}^{-3}{\text{m}}^2}{1.4\Omega }\frac{dB}{dt} \\ =\left(-6.7\times {10}^{-3}{\text{m}}^2{\Omega }^{-1}\right)\frac{dB}{dt} \end{gathered}$$

Consider from the given diagram and the equation of the current the values as:

SN	Time intervals	Time taken	Change in magnetic field density	Rate of change of density	Current I
1	0 to 3	3	0	0	0
2	3 to 5	2	1.6	1.6	$-5.4\times {10}^{-3}$
3	5 to 12	7	0	0	0
4	12 to 20	8	-1.6	-1.6	$+1.3\times {10}^{-3}$

Consider the graph for the variation between time and the current is shown in Figure below.