A spherical capacitor consists of a spherical conducting shell of radius b and charge -Q concentric with a smaller conducting sphere of radius a and charge +Q . Enter an expression for the capacitance of this device in terms of Coulomb's Constnat K_e (just type "k") and the two radii a and b. HINT: you may need to apply a negative sign to your final expression to be sure the capacitance comes out positive! Look at your answer and think about which radii is larger

http://www.univsul.org/Dosekan_Wanekan_D/Third%20Examination%20and%20its%20Solutions.pdf

The capacitance of a spherical capacitor can be calculated using the formula:

C = 4π * K_e * (a * b) / (b - a)

where:
- C is the capacitance of the capacitor,
- K_e is Coulomb's constant,
- a is the radius of the smaller conducting sphere,
- b is the radius of the larger conducting shell.

In this case, since the charge on the larger shell is -Q and the charge on the smaller sphere is +Q, the negative sign in the final expression is required to ensure the capacitance comes out positive.

To derive an expression for the capacitance of a spherical capacitor, we can use the formula for capacitance:

C = Q / V

where C is the capacitance, Q is the charge on one of the conductors, and V is the potential difference between the conductors.

In this case, the charge on the smaller conducting sphere is +Q, while the charge on the larger conducting shell is -Q.

To find the potential difference between the conductors, we consider a Gaussian surface between the two conductors. Since the electric field is zero inside the conductors, the potential difference between the conductors depends only on the charges.

The potential difference is given by:

V = k * (Q / a) - k * (-Q / b)

where k is Coulomb's constant.

Simplifying the expression:

V = k * (Q / a + Q / b)

Now, we can substitute this expression for V into the formula for capacitance:

C = Q / (k * (Q / a + Q / b))

Simplifying further:

C = a * b / (a + b)

Finally, we apply a negative sign to the expression to ensure that the capacitance comes out positive:

C = -a * b / (a + b)

Therefore, the expression for the capacitance of a spherical capacitor in terms of Coulomb's constant (k) and the two radii (a and b) is:

C = -a * b / (a + b)