Q26P - Introduction to Electrodynamics

Question

Calculate the Laplacian of the following functions:

(a) 

(b)

(c) .

(d)

Step-by-Step Solution

Answer

1. The value of is 2.

2. The value of is .

1. The value of  is 0.

2. The value of is .

Step 1: Define the laplacian

The divergence of gradient is called as Laplacian of a vector.The vector  is defined as . theoperator is defined as . The value of Laplacian is obtained as

Here, is the second ordered partial derivative of function.

Step 2: Compute

(a)

To compute an expression substitute the vectors and other required expression and then simplify

The vector  is defined as . The Laplacian of the vector o is defined as. 

Substitute for into.

Thus the value of  is 2.

Step 3: Compute

(b)

To compute an expression substitute the vectors and other required expression and then simplify.

The vector  is defined as . The Laplacian of the vector  is defined as . 

Substitute for into .

Solve further as,

Thus, the value of is .

Step 4: Compute

(c)

To compute an expression substitute the vectors and other required expression and then simplify.

The vector  is defined as . The Laplacian of the vector  is defined as . 

Substitute for  into.

Solve further as,

Thus, the value of  is 0.

Step 5: Compute

(d)

To compute an expression substitute the vectors and other required expression and then simplify.

The vector  is defined as . The Laplacian of the vector  is defined as . 

Substitute for into .

Solve further as,

Thus the value of is.