Electrostatic energy, conductors and dipoles

Problem:

A charge Q is uniformly distributed through the volume of a sphere of radius R.  Calculate the electrostatic energy stored in the resulting electric field.

Problem:

(a)  A positive charge Q is spread over an semicircular arc with radius R as shown.
(1)  What is the work required to bring in a charge -q from infinity to the center of the arc?
(2)  Calculate the magnitude and direction of the force on a charge –q at the center of the arc.

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(b)  The potential of a uniformly charged spherical shell of radius R centered at the origin is
V(r) = q/(4πε0r) r ≥ R,   V(r) = q/(4πε0R) r < R, 
where q denotes the total charge of the sphere.
Calculate the energy that it requires to deposit a charge Q on an initially neutral conducting spherical shell with radius R.  Use two different approaches to come to the result.
(3)  Calculate the energy by incrementally adding a charge dq to the sphere.
(4)  Obtain the energy by considering the resulting electric field of the spherical shell.

Solution:

Problem:

(a) Show that the plates of an isolated parallel-plate capacitor attract each other with a force F = (Q2/2ε0A), where ±Q are the charges on the plates and A is the area of each plate.
[HINT: Consider the work necessary to increase the plate separation from x to x + dx.]
(b) How does your answer to part (a) change if, instead, one maintains a constant potential difference between the plates of the capacitor?

Solution:

Problem:

Assume you have two capacitors of capacitance C and ½C respectively, each holding a charge Q0.  At t = 0 you connect then with wires as shown.

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(a)  What is the energy stored in the capacitors at t = 0?
(b)  What is the energy stored in the capacitors at t --> infinity?
(c)  Explain the difference!

Solution:

Problem:

Three identical capacitors (capacitance C) are connected in series to a source of electric potential V.   The capacitors are then individually disconnected from the source and wired in a new, series-parallel circuit in which two capacitors remain in series in their original orientation and the third is placed in parallel with the first two, with its positive plate connected to the positive end of the series pair, as shown in the figure.  Find the new potential difference, V', of the combined capacitors. 

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Solution:

Problem:

A battery with voltage V charges a capacitor with capacitance C.  At t = 0 the battery is disconnected.  The positive plate of the capacitor is then connected to one plate of a previously uncharged identical capacitor by wire with zero resistance.  The negative plate of the charged capacitor is connected to the other plate of the second capacitor.
(a)  What is the energy stored in the capacitors at t = 0?
(b)  What is the energy stored in the capacitors at t --> infinity?
(c)  Explain the difference from the point of view of the energy conservation.

Solution:

Problem:

Two isolated spherical conductors of radii 3 cm and 9 cm are charged to 1500 V and 3000 V, respectively.  They are very far away from each other.
(a) What is the total energy of the system?
(b) If we connect the two conductors by a fine wire and wait until equilibrium is established, how much heat will be generated in the wire?

Solution:

Problem:

Two identical metal plates of area A each are arranged as shown: the top plate is suspended by a spring with a force constant k and the bottom plate is mounted underneath so that it does not move.  When both plates are neutral, the equilibrium distance between them is h.  If the plates are connected to a power supply, what maximum voltage between the plates would not cause them to touch and maintain a stable equilibrium?

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Solution

Problem:

Prove that the interaction energy of two dipoles P1 and P2 separated by a relative position R is given by
P1∙P2/R3 - 3(P1∙R)(P2∙R)/R5.  (Gaussian units)

Solution: