A wire of radius one meter is centered on the origin in the xy plane. This wire carries a total charge of +Q. Another wire of length two meters runs along the z-axis and is centered at the origin. It carries total charge -Q. What is the electric field vector at any point on the surface of a sphere centered at the origin with radius two meters
To find the electric field at any point on the surface of the sphere, you can use Gauss's Law. Gauss's Law states that the flux of the electric field through a closed surface is related to the charge enclosed by that surface.
In this case, the closed surface we will use is the surface of the sphere centered at the origin with a radius of two meters. Let's call this sphere S.
Since the sphere is centered at the origin, both wires lie on the x-y plane, and their charges are symmetrically distributed along the z-axis. As a result, the electric field due to the wire carrying charge +Q will have a radial component pointing outwards from the origin, while the electric field due to the wire carrying charge -Q will have a radial component pointing inwards towards the origin.
The electric field on the surface of the sphere will be the sum of the electric fields generated by the two wires at that point. The radial component of the electric field due to the wire carrying charge +Q will be given by:
E1 = (k * Q) / (r1^2), where k is the Coulomb's constant, Q is the charge on the wire, and r1 is the distance from the wire to the point on the surface of the sphere.
Similarly, the radial component of the electric field due to the wire carrying charge -Q will be given by:
E2 = (-k * Q) / (r2^2), where r2 is the distance from the wire to the point on the surface of the sphere.
Since the electric field generated by the wire carrying charge +Q points outwards, and the electric field generated by the wire carrying charge -Q points inwards, their sum will give us the net electric field at that point on the surface of the sphere.
Therefore, the net electric field at any point on the surface of the sphere is:
E_net = E1 + E2 = (k * Q) / (r1^2) - (k * Q) / (r2^2)
Substituting the values for the variables, with r1 = 2 meters (the radius of the sphere) and r2 = 2 meters (the length of the wire along the z-axis), we get:
E_net = (k * Q) / (2^2) - (k * Q) / (2^2)
Simplifying this expression:
E_net = (k * Q) / 4 - (k * Q) / 4
E_net = 0
Therefore, the net electric field at any point on the surface of the sphere is zero.