Problem

A long solenoid with 15 turns per cm has a small loop of area $2.0 \, \mathrm{cm}^2$ placed inside the solenoid, normal to its axis. If the current carried by the solenoid changes steadily from $2.0 \, \mathrm{A}$ to $4.0 \, \mathrm{A}$ in $0.1 \, \mathrm{s}$, what is the induced e.m.f. in the loop while the current is changing?

Data

• Number of turns per cm: $15 \, \mathrm{turns/cm} = 1500 \, \mathrm{turns/m}$
• Turns per unit length: $n = 1500 \, \mathrm{turns/m}$
• Area of the loop:

$$A = 2.0 \, \mathrm{cm}^2 = 2.0 \times (10^{-2})^2 \, \mathrm{m}^2 = 2.0 \times 10^{-4} \, \mathrm{m}^2$$

• Initial current: $I_i = 2.0 \, \mathrm{A}$
• Final current: $I_f = 4.0 \, \mathrm{A}$
• Time interval: $\Delta t = 0.1 \, \mathrm{s}$

To find:

• Induced e.m.f. $\varepsilon$.

Prerequisite Concepts

1. Magnetic field inside a solenoid:

$$B = \mu_0 n I$$

where:

• $\mu_0 = 4\pi \times 10^{-7} \, \mathrm{T \cdot m/A}$ is the permeability of free space,
• $n$ is the number of turns per unit length,
• $I$ is the current.

2. Change in magnetic field:

$$\Delta B = \mu_0 n \Delta I$$

3. Faraday’s law of electromagnetic induction:

$$\varepsilon = \frac{\Delta \Phi}{\Delta t}$$

where:

$$\Delta \Phi = \Delta B \cdot A$$

Solution

Step 1: Calculate the Change in Magnetic Field ($\Delta B$)

1. Use the formula:

$$\Delta B = \mu_0 n \Delta I$$

2. Substitute the values:

$$\Delta B = (4\pi \times 10^{-7}) \cdot 1500 \cdot (4 - 2)$$

3. Simplify:

$$\Delta B = 4 \times 3.14 \times 10^{-7} \cdot 1500 \cdot 2$$ $$\Delta B = 3.77 \times 10^{-4} \, \mathrm{T}.$$

Step 2: Calculate the Induced e.m.f. ($\varepsilon$)

1. Use Faraday’s law:

$$\varepsilon = \frac{\Delta \Phi}{\Delta t} = \frac{\Delta B \cdot A}{\Delta t}$$

2. Substitute the values:

$$\varepsilon = \frac{(3.77 \times 10^{-4}) \cdot (2.0 \times 10^{-4})}{0.1}$$

3. Simplify:

$$\varepsilon = \frac{7.54 \times 10^{-8}}{0.1} = 7.54 \times 10^{-5} \, \mathrm{V}.$$

Answer

The induced e.m.f. in the loop is:

$$\varepsilon = 7.54 \times 10^{-5} \, \mathrm{V}.$$

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