Calculating the Electric Field at the Origin

To calculate the electric field at the origin, we need to use the formula:

E = -dV/dx

where E is the electric field, V is the electric potential, and x is the position along the x-axis.

Finding the Electric Potential Gradient

From the problem statement, we know that the electric potential decreases uniformly from 60 V to 20 V between x=-2 to x=2 m. This means that the electric potential gradient is constant:

dV/dx = (20 V - 60 V) / (2 m - (-2 m))
dV/dx = -10 V/m

Calculating the Electric Field at the Origin

To find the electric field at the origin, we need to evaluate the above formula at x=0:

E = -dV/dx @ x=0
E = -(-10 V/m)
E = 10 V/m

Therefore, the magnitude of the electric field at the origin is 10 V/m.

Explanation

The electric field is a vector quantity that describes the force exerted on a charged particle at a given point in space. It is related to the electric potential by the equation E = -dV/dx, which states that the electric field is the negative gradient of the electric potential. In this problem, we were given the electric potential as a function of position along the x-axis and asked to find the electric field at the origin. To do this, we found the electric potential gradient, which was constant over the given interval, and then evaluated the electric field formula at x=0. The resulting magnitude of the electric field was 10 V/m.

-10 V/m