Question

A charge Q is uniformly distributed along the x axis from x = a to x = b. If Q = 45

nC, a = -3.0 m, and b = 2.0 m, what is the electric potential (relative to zero at infinity)

at the point, x=8.0 m, on the axis?​

Answer 1

The potential at the point 8 meters is approximately 48.98 Joules/C

Step-by-step explanation:

Notice that we need to find the potential at a point aligned to the rod's axis (see attached figure), that is located 6 meters from one end of the rod. Notice as well, that the length (L) of the rod is 5 meters.

Since we have a uniformly distributed charge, the charge density per unit of length is defined as:

`\lambda=(Q)/(L) \\\lambda=(45)/(5)\,(10^(-9)\,C)/(m) \\\lambda=9\,(10^(-9)\,C)/(m)`

In order to find the contribution of each little segment dx of charge

`dq=\lambda\,dx`

to the potential at the requested point, we need to perform an integral:

`V=\int\limits^(11)_(6) {} \, dV \\V=\int\limits^(11)_(6) {} \, k\,(dq)/(x) \\V=\lambda \,*k\,\int\limits^(11)_(6) {} \, (dx)/(x) \\V=8.98*9 \, \,ln((11)/(6)) \,(J)/(C) \\V=80.82\,* \,0.606 \,(J)/(C)\\V=48.98 (J)/(C)`