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

Question

The figure (Figure 1) shows a thin rod of length LL with total charge QQ.

Verify that your expression has the expected behavior if r >> L.

r≫L.

 

Answer 1

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 .

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

That is .

Thus, .

Solution:

.