Find an expression for the electric field strength on the axis of the rod at distance r from the center.

Question

A)Find an expression for the electric field strength on the axis of the rod at distance rr from the center.  E=

B)Verify that your expression has the expected behavior if r >>L. E=

C)Evaluate E at r= 4.7 cm if L=5.0 cm and Q=3.8nC. E=

 

   E=

Answer 1

(A)

Step 1:

Define a coordinate system along the axis of the rod with the origin at the center of the rod as shown.

Divide the rod into small pieces of charge  of length  at the location  on the rod.

Electric field strength at a point  due to a piece of rod of charge  is  .

Where  is the position vector of point .

Position vector .

.

As the electric field is having only  component and there is no  components.

Substitute  and solve electric field for only  component.

Therefore .

Here Charge is positive and  is in  direction.

As the rod is uniformly charged, charge of piece divide by its length is same as the  total charge divide by total length of the rod.

That is .

Step 2:

By super position theorem, Net field strength at point  is summation of field strength due to each piece of rod, which can be found by integrating  under the limits and .

.

Solution:

(A) .

Answer 2

(B)

Consider the point  is very far away from the rod, then rod length is negligible. 

That is .

Thus, .

Answer 3

(C)

Distance from the center of the rod to point is .

Length of the rod is .

Point charge is .

Elcetric field intensity .

 

Substitute corresponding values in the formula.

.