Find the general solution to Laplace's equation (∇²V = 0) for V(r) in spherical coordinates.

You're on a roll — here's the solution to question 1: We need to find the general solution to Laplace's equation (^2 V = 0) for two cases where the potential V depends only on the radial coordinate r. Case 1: Spherical Coordinates Step 1: Write Laplace's equation in spherical coordinates. Laplace's equation in spherical coordinates (r, , ) is: ^2 V = (1)/(r^2) ()/( r) ( r^2 ( V)/( r) ) + (1)/(r^2 ) ()/( ) ( ( V)/( ) ) + (1)/(r^2 ^2) (^2 V)/( ^2) = 0 Step 2: Simplify the equation for V(r). Given that V depends only on r, the partial derivatives with respect to and are zero (( V)/( ) = 0 and ( V)/( ) = 0). The equation simplifies to: (1)/(r^2) ()/( r) ( r^2 ( V)/( r) ) = 0 Step 3: Integrate the simplified equation. Multiply by r^2 and integrate with respect to r: ()/( r) ( r^2 ( V)/( r) ) = 0 r^2 ( V)/( r) = A where A is an integration constant. Step 4: Integrate again to find V(r). ( V)/( r) = (A)/(r^2) V(r) = (A)/(r^2) dr = A r^-2 dr = A ( -(1)/(r) ) + B Let C_1 = -A and C_2 = B. The general solution in spherical coordinates for V(r) is: V(r) = (C_1)/(r) + C_2 Case 2: Cylindrical Coordinates Step 1: Write Laplace's equation in cylindrical coordinates. Laplace's equation in cylindrical coordinates (r, , z) is: ^2 V = (1)/(r) ()/( r) ( r ( V)/( r) ) + (1)/(r^2) (^2 V)/( ^2) + (^2 V)/( z^2) = 0 Step 2: Simplify the equation for V(r). Given that V depends only on r, the partial derivatives with respect to and z are zero (( V)/( ) = 0 and ( V)/( z) = 0). The equation simplifies to: (1)/(r) ()/( r) ( r ( V)/( r) ) = 0 Step 3: Integrate the simplified equation. Multiply by r and integrate with respect to r: ()/( r) ( r ( V)/( r) ) = 0 r ( V)/( r) = A where A is an integration constant. Step 4: Integrate again to find V(r). ( V)/( r) = (A)/(r) V(r) = (A)/(r) dr = A (r) + B Let C_1 = A and C_2 = B. The general solution in cylindrical coordinates for V(r) is: V(r) = C_1 (r) + C_2 What's next?