In this video, we compute the electric field of a uniformly charged rod, where we measure the electric field at a point that lies on the axis of the rod (the electric field at an axial point).
We begin by breaking the charged rod into point charges - these are infinitesimal line segments of width dx containing a charge of dq.
Next, we find an expression for the distance from a point charge at a location of x to the observation point where we are measuring the electric field.
After a quick reminder on the definition of linear charge density, we can express each dq as lambda*dx, and we're ready to express the electric field contribution due to dq as an expression in terms of x.
Once the field contribution of dq is set up, we can set up the electric field integral for a uniform line of charge, and we integrate as x goes from 0 to L to cover the entire charged rod. We obtain a formula for electric field of a uniformly charged rod, and we simplify to a single fraction.
Finally, we take the large distance limit for our expression of electric field: when the distance becomes large compared to the length of the charged rod, we retrieve our usual expression for the electric field of a point charge in the large distance limit.
