Problem 31 - Master Resource Book in JEE Main Physics

Question

What is self-inductance of a coil which produces \(5 \mathrm{~V}\), when current in it changes from \(3 \mathrm{~A}\) to \(2 \mathrm{~A}\) in one millisecond? (a) \(5000 \mathrm{H}\) (b) \(5 \mathrm{mH}\) (c) \(50 \mathrm{H}\) (d) \(5 \mathrm{H}\)

Step-by-Step Solution

Answer

The self-inductance of the coil is 5 mH.

Step 1: Understanding the Problem

We are asked to find the self-inductance of a coil given that it produces an electromotive force (EMF) of 5 V when the current changes from 3 A to 2 A in 1 millisecond. We need to remember that the formula for EMF induced in a coil due to self-inductance is given by: \[ \varepsilon = -L \frac{\Delta I}{\Delta t} \] where \( \varepsilon \) is the induced EMF, \( L \) is the self-inductance, \( \Delta I \) is the change in current, and \( \Delta t \) is the change in time.

Step 2: Calculating Change in Current and Time

The change in current \( \Delta I \) is computed by subtracting the final current from the initial current: \[ \Delta I = 3 \text{ A} - 2 \text{ A} = 1 \text{ A} \]The change in time \( \Delta t \) is given as 1 millisecond, which we convert to seconds: \[ \Delta t = 1 \text{ ms} = 0.001 \text{ s} \].

Step 3: Using the Formula for Induced EMF

We substitute the known values into the formula to solve for the self-inductance \( L \):\[ \varepsilon = -L \frac{\Delta I}{\Delta t} \].Given \( \varepsilon = 5 \text{ V} \), \( \Delta I = 1 \text{ A} \), and \( \Delta t = 0.001 \text{ s} \), we rearrange the formula to solve for \( L \):\[ 5 = -L \times \frac{1}{0.001} \].

Step 4: Solving for Self-Inductance

Rearrange and solve the equation to find \( L \):\[ L = -\frac{5}{-1000} = \frac{5}{1000} = 0.005 \text{ H} \]. Convert 0.005 H to millihenries (mH): \[ 0.005 \text{ H} = 5 \text{ mH} \].

Step 5: Conclusion

Based on the calculated self-inductance value, the correct answer is \( 5 \text{ mH} \), which matches option (b).

Key Concepts

Induced EMF

Electromotive force (EMF) refers to the voltage generated when there is a change in magnetic flux through a circuit, such as a coil. In the context of self-inductance, an induced EMF is generated due to the coil's own changing current. This phenomenon is governed by Lenz's Law, which ensures that the direction of the induced EMF opposes the change in current causing it. In simple terms, if you increase the current in the coil, the induced EMF will try to decrease it, and vice versa.
The relationship between the induced EMF \( \varepsilon \) and self-inductance \( L \) can be expressed mathematically as:

• \( \varepsilon = -L \frac{\Delta I}{\Delta t} \)

With this formula, you can determine the magnitude of voltage (EMF) induced when certain changes in current occur over time through a coil. This allows understanding how some electrical components react to changes dynamically!

Change in Current

When discussing self-inductance, one crucial aspect to consider is the change in current, denoted as \( \Delta I \). This change is critical because it directly influences the induced EMF in the coil. Simply stated, \( \Delta I \) is the difference between the final and initial current flowing through the coil.
For example, in the given problem, the current drops from 3 A to 2 A. This results in a change in current of 1 A. The straightforward calculation can be expressed as:

• \( \Delta I = 3 \text{ A} - 2 \text{ A} = 1 \text{ A} \)

The larger the change in current, the larger the induced EMF, provided the change occurs within the same time frame. Thus, monitoring the current variations is essential when calculating or predicting outcomes related to self-inductance.

Change in Time

The change in time \( \Delta t \) is another vital factor in determining the induced EMF in a coil. Here, it refers to the duration over which the current changes occur. It's imperative to convert this time measurement into seconds when using it in electromagnetic formulas for consistency with standard units.
In our example problem, the time change is given as 1 millisecond, which translates into:

• \( \Delta t = 1 \text{ ms} = 0.001 \text{ s} \)

A faster change in current over a shorter time increases the magnitude of the induced EMF since \( \varepsilon \propto \frac{1}{\Delta t} \). Understanding this relation means quicker current changes can lead to more significant electromagnetic reactions within the coil.