SOLUTION

To determine the new power factor of an RL circuit when the frequency is doubled, we can follow these steps:

Given Information

Initial power factor $ \cos \phi = \frac{1}{\sqrt{2}}$

Step 1: Understanding the Power Factor

The power factor for an RL circuit is given by:

$\cos \phi = \frac{R}{Z}$

where:

R is the resistance,
Z is the impedance.

The impedance Z  for an RL circuit is given by:

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

where X_L is the inductive reactance, which is defined as:

X_L = ω_L

ω = 2πf

where f  is the frequency

Step 2: Initial Conditions

From the initial power factor:

$\cos \phi = \frac{R}{Z} = \frac{1}{\sqrt{2}}$

This implies that:

Z = R √2

Using the impedance formula:

$Z^2 = R^2 + (X_L)^2$

Substituting for Z :

$(R \sqrt{2})^2 = R^2 + (X_L)^2$

$2R^2 = R^2 + (X_L)^2$

$(X_L)^2 = R^2$

Thus, we find that:

X_L = R

Step 3: Doubling the Frequency

When the frequency is doubled, the new inductive reactance  X_L = R becomes:

$X_L’ = 2X_L = 2R$

Step 4: New Impedance Calculation

Now, we calculate the new impedance  Z’:

$Z’ = \sqrt{R^2 + (X_L’)^2} = \sqrt{R^2 + (2R)^2} = \sqrt{R^2 + 4R^2} = \sqrt{5R^2} = R\sqrt{5}$

Step 5: New Power Factor Calculation

Now we can find the new power factor $\cos \phi’$:

$\cos \phi’ = \frac{R}{Z’} = \frac{R}{R\sqrt{5}} = \frac{1}{\sqrt{5}}$

Conclusion

Thus, when the frequency is doubled, the new power factor of the RL circuit becomes:

$\frac{1}{\sqrt{5}}$