Question

A student connects a long air cored - coil of manganin wire to a 100 V D.C. supply and records a current of 25 amp. When the same coil is connected across 100 V. 50 Hz a.c. the current reduces to 20 A, the reactance of the coil is :-

• A. 4 $\Omega$
• B. 3 $\Omega$
• C. 5 $\Omega$
• D. None

Answer 1

Answer: (B)

3 $\Omega$

Answer 2

The problem involves analyzing the behavior of a coil in both DC and AC circuits to determine its reactance.

1. Determine the Resistance (R) of the Coil (DC Circuit):

When the coil is connected to a DC supply, there is no inductive reactance ($X_L = 2\pi f L = 0$ since $f=0$ for DC). The current is limited solely by the resistance of the coil.

Given:

DC Voltage ($V_{DC}$) = 100 V DC Current ($I_{DC}$) = 25 A

Using Ohm's Law for DC:

$R = \frac{V_{DC}}{I_{DC}}$

$R = \frac{100\,V}{25\,A}$

$R = 4\,\Omega$

2. Determine the Impedance (Z) of the Coil (AC Circuit):

When the same coil is connected to an AC supply, the current is limited by its impedance (Z), which is the combined effect of resistance (R) and inductive reactance ($X_L$).

Given:

AC Voltage ($V_{AC}$) = 100 V AC Current ($I_{AC}$) = 20 A

Using Ohm's Law for AC:

$Z = \frac{V_{AC}}{I_{AC}}$

$Z = \frac{100\,V}{20\,A}$

$Z = 5\,\Omega$

3. Calculate the Reactance ($X_L$) of the Coil:

For a coil (which is essentially an inductor with resistance), the impedance (Z) is related to resistance (R) and inductive reactance ($X_L$) by the formula:

$Z = \sqrt{R^2 + X_L^2}$

We need to find $X_L$. Rearranging the formula:

$X_L^2 = Z^2 - R^2$

$X_L = \sqrt{Z^2 - R^2}$

Substitute the values of Z and R we calculated:

$X_L = \sqrt{(5\,\Omega)^2 - (4\,\Omega)^2}$

$X_L = \sqrt{25\,\Omega^2 - 16\,\Omega^2}$

$X_L = \sqrt{9\,\Omega^2}$

$X_L = 3\,\Omega$

The reactance of the coil is $3\,\Omega$.