Problem

A circuit with 1000 turns encloses a magnetic circuit of cross-sectional area $20 \, \mathrm{cm}^2$. When a current of $4 \, \mathrm{A}$ flows, the flux density is $1 \, \mathrm{Wb/m}^2$, and for $9 \, \mathrm{A}$, the flux density is $1.4 \, \mathrm{Wb/m}^2$. Find:
(a) The mean value of the inductance between these current limits.
(b) The induced e.m.f. if the current falls from $9 \, \mathrm{A}$ to $4 \, \mathrm{A}$ in $0.05 \, \mathrm{s}$.

Data

• Number of turns: $N = 1000$
• Area of cross-section:

$$A = 20 \, \mathrm{cm}^2 = 20 \times 10^{-4} \, \mathrm{m}^2$$

• Initial magnetic flux density: $B_i = 1 \, \mathrm{Wb/m}^2$
• Final magnetic flux density: $B_f = 1.4 \, \mathrm{Wb/m}^2$
• Initial current: $I_i = 4 \, \mathrm{A}$
• Final current: $I_f = 9 \, \mathrm{A}$
• Time interval: $\Delta t = 0.05 \, \mathrm{s}$

To find:

1. Self-inductance $L$.
2. Induced e.m.f. $\varepsilon$.

Prerequisite Concepts

1. Self-inductance:

$$L = \frac{N \Delta \Phi}{\Delta I},$$

where $\Delta \Phi$ is the change in magnetic flux.

2. Magnetic flux:

$$\Phi = B \cdot A,$$

and change in flux:

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

3. Induced e.m.f.:

$$\varepsilon = L \frac{\Delta I}{\Delta t}.$$

Solution

Step 1: Self-Inductance

1. Substitute the flux change formula into the inductance equation:

$$L = \frac{N A \Delta B}{\Delta I}.$$

2. Calculate $\Delta B$ and $\Delta I$:

$$\Delta B = B_f - B_i = 1.4 - 1.0 = 0.4 \, \mathrm{Wb/m}^2,$$ $$\Delta I = I_f - I_i = 9 - 4 = 5 \, \mathrm{A}.$$

3. Substitute values:

$$L = \frac{1000 \cdot (20 \times 10^{-4}) \cdot 0.4}{5}.$$

4. Simplify:

$$L = \frac{1000 \cdot 8 \times 10^{-3}}{5} = 0.16 \, \mathrm{H}.$$

Step 2: Induced e.m.f.

1. Use the e.m.f. formula:

$$\varepsilon = L \frac{\Delta I}{\Delta t}.$$

2. Substitute values:

$$\varepsilon = 0.16 \cdot \frac{9 - 4}{0.05}.$$

3. Simplify:

$$\varepsilon = 0.16 \cdot \frac{5}{0.05} = 0.16 \cdot 100 = 16 \, \mathrm{V}.$$

Answer

1. Mean self-inductance:

$$L = 0.16 \, \mathrm{H}.$$

2. Induced e.m.f.:

$$\varepsilon = 16 \, \mathrm{V}.$$

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