Question

Two coils of wire are near each other, positioned on a common axis (Figure 22.57). Coil 1 is connected to a power supply whose output voltage can be adjusted by turning a knob so that the current ${\mathit{I}}_{\mathbf{1}}$in coil 1 can be varied, and ${\mathit{I}}_{\mathbf{1}}$is measured be ammeter 1.

Current ${\mathit{I}}_{\mathbf{2}}$in coil 2 is measured by ammeter 2. The ammeters have needles that deflect positive or negative depending on the direction of current passing through the ammeter, and ammeters read positive if conventional current flows into the + terminal. Figure 22.58 is a graph of ${\mathit{I}}_{\mathbf{1}}$vs. time. Draw a graph of ${\mathit{I}}_{\mathbf{2}}$vs. time over the same time interval. Explain your reasoning.

Short answer

From first graph it is clear that the slope of the current $\left(I_1\right)$curve of coil 1 is for the time interval $t=0\text{sto}t=4\text{s}$is less than the slope of the current $\left(I_1\right)$curve of coil 1 is for the time interval $t=6\text{sto}t=8\text{s}$. So the induced current $\left(I_2\right)$in the coil 2 for the time interval $t=0\text{sto}t=4\text{s}$is greater than the current $\left(I_2\right)$ curve of coil 2 is for the time $t=6\text{sto}t=8\text{s}$ .

Step-by-step solution

A concept:

Faraday's law states that a changing magnetic field will create a curling electric field. If we put a loop of wire nearby, we can add up little bits of length along that loop with a curly electric field, which tells us how curly that electric field is.

A given data:

Two coils of wire are near each other, positioned on a common axis (Figure 22.57). Coil 1 is connected to a power supply whose output voltage can be adjusted by turning a knob so that the current $I_1$in coil 1 can be varied, and $I_1$is measured be ammeter 1.

Current $I_2$in coil 2 is measured by ammeter 2. The ammeters have needles that deflect positive or negative depending on the direction of current passing through the ammeter, and ammeters read positive if conventional current flows into the $+$terminal.

Draw a graph of  I2 vs. time over the same time interval. Explain your reasoning:

Current $\left(I_1\right)$in the coil 1 is increasing from the current $0Ato3A$in time $4s$(this current runs in the clockwise direction) and the magnetic field inside coil 1 increasing.

So from Faraday’s law, the current in coil 2 runs in counter inside coil 1. Thus the ammeter reads negative current.

Current $\left(I_1\right)$in the coil 1 is constant from time $4sto6s$and magnetic field inside the coil 1 constant.

Current $\left(I_1\right)$in the coil 1 is decreasing from time $3Ato0A$in time $6sto8s$(this current runs in clock wise direction) and magnetic field inside the coil 1 decreasing.

So, from Faraday’s law, the current in coil 2 runs in a clock direction to increase the magnetic field inside coil 1. Thus, the ammeter reads a positive current.

Above situation is shown in the below graph.

The conclusion:

From first graph it is clear that the slope of the current $\left(I_1\right)$curve of coil 1 is for the time interval $t=0\text{sto}t=4\text{s}$is less than the slope of the current $\left(I_1\right)$curve of coil 1 is for the time interval $t=6\text{sto}t=8\text{s}$. So the induced current $\left(I_2\right)$ in the coil 2 for the time interval $t=0\text{sto}t=4\text{s}$is greater than the current $\left(I_2\right)$ curve of coil 2 is for the time $t=6\text{sto}t=8\text{s}$ .