Problem 35 - Master Resource Book in JEE Main Physics

Question

A coil of wire of certain radius has 100 turns and a self-inductance of \(15 \mathrm{mH}\). The self-inductance of a second similar coil of 500 turns will be (a) \(75 \mathrm{mH}\) (b) \(375 \mathrm{mH}\) (c) \(15 \mathrm{mH}\) (d) None of these

Step-by-Step Solution

Answer

The self-inductance of the second coil is 375 mH, option (b).

Step 1: Understanding Self-Inductance

Self-inductance, denoted by \( L \), of a coil is a measure of how much voltage is induced in the coil when the current through it changes. It depends on the square of the number of turns in the coil.

Step 2: Using the Formula for Self-Inductance

For a coil, the self-inductance \( L \) is proportional to the square of the number of turns \( N \) and can be expressed by the relationship \( L \propto N^2 \). If we have an initial coil with \( N_1 = 100 \) turns and \( L_1 = 15 \mathrm{mH} \), and a second similar coil with \( N_2 = 500 \) turns, we can use the proportional relationship to determine \( L_2 \).

Step 3: Setting Up Proportional Equation

The relationship between the self-inductances of the two coils can be set up as \( \frac{L_2}{L_1} = \left( \frac{N_2}{N_1} \right)^2 \). Here, \( N_1 = 100 \) and \( N_2 = 500 \). Substituting the known values, the equation becomes \( \frac{L_2}{15} = \left( \frac{500}{100} \right)^2 \).

Step 4: Calculating the Ratio

Calculate \( \left( \frac{500}{100} \right)^2 = \left( 5 \right)^2 = 25 \). So, the equation becomes \( \frac{L_2}{15} = 25 \).

Step 5: Solving for L2

To find \( L_2 \), multiply both sides of the equation by 15: \( L_2 = 25 \times 15 \).

Step 6: Final Calculation

Calculate \( 25 \times 15 = 375 \). Therefore, the self-inductance \( L_2 = 375 \mathrm{mH} \).

Key Concepts

Number of Turns in a Coil

The number of turns in a coil, often denoted as \( N \), plays a crucial role in determining the coil's electrical properties. Each turn of the coil contributes to its ability to produce magnetic fields and store energy as the coil is energized. A higher number of turns results in a stronger magnetic field, enhancing the coil's inductive properties.

• More turns lead to increased self-inductance, making the coil more effective at storing magnetic energy.
• Coils with more turns will typically have higher resistance due to the longer length of wire required.

For example, in this exercise, we learned that a coil with 100 turns has a certain self-inductance. But when the number of turns is increased to 500 in a similar coil, the self-inductance becomes significantly higher, demonstrating the direct relationship between the number of turns and self-inductance.

Proportional Relationships

Proportional relationships are an essential part of understanding how different quantities interact in physics. When one quantity changes, another can change in a predictable manner according to a set rule.
The concept of proportionality is key when calculating self-inductance in coils. The self-inductance \( L \) is proportional to the square of the number of turns \( N \): \( L \propto N^2 \). This means:

• If you double the number of turns, the self-inductance will increase by a factor of four (since \((2N)^2 = 4N^2\)).
• If the number of turns is increased five times (as in our 100 to 500 turns scenario), the self-inductance will increase by 25 times the original (since \((5N)^2 = 25N^2\)).

Understanding proportional relationships allows you to predict how changes in one aspect of a system affect another, which is critical when designing and using inductive components.

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a coil induces a voltage (or electromotive force). This principle underpins the operation of many electrical devices, such as transformers and inductors, and is also the basis for self-inductance.

• It occurs when the magnetic field around a conductor varies, either by changing the current passing through the conductor or by moving the conductor within the magnetic field.
• Induced voltage depends on the rate of change of the magnetic field and the number of turns in the coil.

The exercise demonstrates electromagnetic induction through the concept of self-inductance, where the voltage induced in a single coil is a result of its own changing magnetic field. Higher self-inductance in the exercise with more turns results in a greater ability to induce voltage, clearly illustrating this fundamental electrical principle.