Question

In circular coil. when no. of turns is doubled \& resistance becomes half of the initial then inductance becomes ...... (a) 4 times (b) 2 times (c) 8 times (d) No change

Short answer

The inductance becomes \(2\) times the initial inductance when the number of turns is doubled and resistance becomes half of the initial.

Step-by-step solution

Determine the initial inductance of the coil

Let's denote the initial number of turns as N, resistance as R, and inductance as L. At first, we haven't made any changes to these values.

Find the formula to connect inductance, number of turns, and resistance

From the formula of the inductor, we know that inductance(L) is proportional to the square of the number of turns (N^2) and the reluctance R (L∝N^2R). So, we can write: L = k × N^2 × R where k is a proportionality constant.

Double the number of turns and halve the resistance

Now, let's double the number of turns and halve the resistance. We denote the new number of turns as N' and the new resistance as R'. Thus, N' = 2N and R' = R/2.

Find the new inductance L'

Using the formula from Step 2, we can find the new inductance L' with the new values of N' and R'. L' = k × (N')^2 × R' Substituting values of N' and R' we get: L' = k × (2N)^2 × (R/2)

Simplifying the equation and finding the ratio of the new inductance to the initial inductance

We can simplify the equation for the new inductance and find the ratio of L' to L. L' = k × 4N^2 × R/2 L' = 2k × N^2 × R Now, let's find L'/L: L'/L = (2k × N^2 × R) / (k × N^2 × R) The factors of k, N^2, and R cancel out: L'/L = 2

Choose the correct answer

From our calculations in Step 5, we've determined that the new inductance L' is 2 times the initial inductance L. So, the correct answer is: (b) 2 times

Key concepts

Number of Turns

The number of turns in a coil is a crucial factor influencing its inductance. Inductance is the ability of a coil to resist changes in electrical current, and it is represented by the symbol L. The relationship between the number of turns in a coil, denoted as N, and the inductance is quadratic. This means if you double the number of turns, the inductance does not just double; instead, it increases fourfold. This is because inductance is proportional to the square of the number of turns, mathematically expressed as:\[ L \propto N^2 \]Understanding this relationship helps in predicting how changes in the coil will affect its inductance. More turns mean more loops for the magnetic field to interact with, thus increasing its inductance significantly.

Resistance in a Coil

Resistance is another vital parameter when considering coils and their inductance. It represents how much the coil resists the flow of electric current through it and is usually denoted by R. Resistance affects inductance indirectly, as a higher resistance means more energy is lost as heat rather than contributing to the inductive properties of the coil.In the context of changing the resistance in a coil, halving the resistance, for example, allows the coil to deliver more of the magnetic effect, thus improving inductance. When resistance is altered, as in the original exercise where it is reduced by half, it forms part of the calculation to determine how new configurations will affect the inductance:\[ L' = k \times N^2 \times \left( \frac{R}{2} \right) \]This change paves the way for adjusting and ultimately enhancing the coil's performance by optimizing both turns and resistance.

Proportionality Constant

The proportionality constant, often denoted by k, plays a key role in linking the mathematical expression to real-world measurements in a coil's inductance. Because real coils do not exist in a vacuum, this constant incorporates various factors such as material properties, coil configuration, and geometrical dimensions.In the formula:\[ L = k \times N^2 \times R \]k serves as a crucial link that helps convert theoretical calculations into practical realizations. Although it doesn't affect the final numerical calculations when comparing ratios such as \[ \frac{L'}{L} \]where k cancels out, appreciating its significance is important in design calculations. It ensures the final inductance calculated considers the real-world features and characteristics of the coil.