Q. 69

Expert-verified
Found in: Page 927

### Physics for Scientists and Engineers: A Strategic Approach with Modern Physics

Book edition 4th
Author(s) Randall D. Knight
Pages 1240 pages
ISBN 9780133942651

# a. Show that the average power loss in a series RLC circuit is b. Prove that the energy dissipation is a maximum at .

(a) The proof of the average power loss in a series RLC circuit is given below.

(b) The proof of the energy dissipation is a maximum at is given below.

See the step by step solution

## Part(a) Step 1: Given information

We have been given that we need to show the average power loss in a series RLC circuit is

## Part(a) Step 2: Proof

We know that the power dissipated () is given by:

(Let this equation be )

where, is root mean square current, is root mean square voltage and is the power factor.

We know the formulas,

and ,

where is the resistance in Ohms () and is impedance.

Substituting the values of and in equation (), we get:

(Let this equation be )

We know the formula to calculate :

,

where is the inductive reactance and is the capacitive reactance.

Substituting and , we get:

Substituting , we get:

Now, substitute the value of in equation to get the expression of .

Hence Proved.

## Part(b) Step 1: Given information

We have been given that we need to prove the energy dissipation is a maximum at .

## Part(b) Step 2: Proof

Energy dissipation is maximum when .

Now simplify the derivative of obtained in Part(a) with respect to :

On simplifying, we get:

From the previous expression, we can observe a relation which is solved by .

Hence, we have proved that the power is maximized when .