Given information:
- Two charged conducting spheres of radii a and b are connected to each other by a wire.

To find:
- The ratio of electric fields at the surfaces of two spheres.

Solution:

Step 1: Understanding the situation
- We have two charged conducting spheres of different radii connected by a wire.
- Since the spheres are conducting, the charges on their surfaces will distribute evenly.
- Let the charge on the first sphere be q1 and the charge on the second sphere be q2.

Step 2: Electric field at the surface of a sphere
- The electric field at the surface of a sphere is given by the equation E = k * (q/r^2), where k is the electrostatic constant, q is the charge on the sphere, and r is the radius of the sphere.
- Therefore, the electric field at the surface of the first sphere is E1 = k * (q1/a^2) and the electric field at the surface of the second sphere is E2 = k * (q2/b^2).

Step 3: Understanding the connection
- Since the two spheres are connected by a wire, they are at the same potential.
- This means that the electric potential at the surface of both spheres is the same.
- The electric potential at the surface of a sphere is given by the equation V = k * (q/r), where V is the electric potential, q is the charge on the sphere, and r is the radius of the sphere.
- Therefore, the electric potential at the surface of the first sphere is V1 = k * (q1/a) and the electric potential at the surface of the second sphere is V2 = k * (q2/b).

Step 4: Ratio of electric fields
- The electric field at a point is the negative gradient of the electric potential at that point.
- Therefore, the ratio of electric fields at the surfaces of the two spheres is given by the equation E1/E2 = (dV1/dr)/(dV2/dr).
- Since the potential is constant at the surface of a sphere, the derivative of the potential with respect to the radius is zero.
- Therefore, the ratio of electric fields at the surfaces of the two spheres is E1/E2 = 0/0, which is an indeterminate form.

Step 5: Applying L'Hopital's Rule
- To evaluate the indeterminate form, we can apply L'Hopital's Rule, which states that if the limit of the ratio of two functions is of the form 0/0 or infinity/infinity, then the limit of the ratio is equal to the limit of the ratio of the derivatives of the two functions.
- Applying L'Hopital's Rule to the ratio E1/E2, we have E1/E2 = (d^2V1/dr^2)/(d^2V2/dr^2).

Step 6: Derivatives of electric potentials
- Taking the derivatives of the electric potentials V1 and V2 with respect to the radius, we have dV1/dr = k * (dq1/dr)/a and dV2/dr = k * (dq2/dr)/b.

Step 7: Ratio of charge derivatives