The electric potential inside a charged spherical conductor of radius R is given by V = kQ/R, and the potential outside is given by V = kQ/r. Using Er = -dV/dr, derive the electric field inside and outside this charge distribution.
To derive the electric field inside and outside the charged spherical conductor, we use the relationship between electric field (E) and electric potential (V). Recall that electric field is the negative gradient of electric potential with respect to distance.
Inside the conductor (r < R):
Given: V = kQ/R
To find the electric field, we need to find the derivative of V with respect to r, which is Er for the electric field along the radial direction.
Taking the derivative of V with respect to r:
dV/dr = -kQ/R²
Since the electric field is Er = -dV/dr, we have:
Er = -(-kQ/R²)
Er = kQ/R²
Therefore, the electric field inside the charged spherical conductor is Er = kQ/R², directed radially inward.
Outside the conductor (r > R):
Given: V = kQ/r
Taking the derivative of V with respect to r:
dV/dr = -kQ/r²
Since the electric field is Er = -dV/dr, we have:
Er = -(-kQ/r²)
Er = kQ/r²
Therefore, the electric field outside the charged spherical conductor is Er = kQ/r², directed radially outward.
In summary:
Inside the conductor (r < R): Er = kQ/R² (inward)
Outside the conductor (r > R): Er = kQ/r² (outward)