Electric potential energy is due to the system of charges.

When we assemble a charge distribution, we need to do some work in assembling the charges. The work done by the external force to assemble the charge is stored in the system of charges as Potential energy of the system of charges.

  • Suppose we have to make an assembly of a few charges. We can do this by putting charges at their respective position one by one.
  • The work done in bringing the first charge Q1 from infinity and placing it at its respective position is zero as there is no other charge present to exert any force on the first charge. Work done for placing the first charge (W1=0) is zero.

W1=0

  • Now when we bring the second charge from infinity in the presence of the first charge, the external force has to do some work against the electrostatic repulsive force of the first charge.

Therefore, W2( work done in placing 2nd charge )= U12(   potential energy  of first and second charge) = KQ1*Q2/ r12

W2= KQ1*Q2/r12, where r12 is the distance between r1 and r2.

  • Now if we wish to bring the third charge Q3 from infinity to its respective position, an external force has to do work against the electrostatic repulsion of both Q1 and Q2.

Therefore work done by the external force in bringing the charge

Q3= W3= U13+U23 = KQ1*Q3/r13 + KQ2*Q3/r23

  • So the Potential energy stored in the system of charge is the total work done in assembling three charges that will be

Potential energy   W= W1+ W2+ W3

= 0 + K*Q1*Q2/r12 + K*Q1*Q3/r13+ K*Q2*Q3/r23

  • Now for a system of four charges, the total potential energy will be the work done to bring all four charges together.

In the above example, I have shown the potential energy for a system of four charges placed at the corner of the square of the side “a” and diagonal “D”.

Electric Potential energy in the external Electric field.

Suppose we have two charges q1  and q2 placed in an external electric field at positions r1 and r2 with respect to some origin. These charges are separated by distance ‘rd’

  •  Now suppose that the electric potential at the position of q1  is V(r1) due to external field E. So work done in placing the charge q1 at r1  in the presence of external field E will be q1(Vr1)
  • The electric potential at the position of q2 is V(r2) due to external field E. So the work done in placing the charge at r2 in the presence of external electric field E is  q2(Vr2)
  • But in the case of the second charge, there will be influence due to charge q1 also. We need to bring q2 in the presence of both E and q1. So work done by the external force in bringing the charge q2 in presence of q1  at a separation ‘rd’ is = Kq1q2/rd

Therefore potential energy due to charges placed in the external electric field will be as shown below.

The potential energy of a dipole in an external electric field.

When we place an electric dipole in an external electric field, the charges will experience an electrostatic force qE. The forces on either charge of the electric dipole are equal in magnitude but opposite in direction.

The direction of the force will be in the direction of E for the positive charge and opposite to E for the negative charge.

Since their magnitudes are the same, these forces cancel each other. So there will be no translational motion in the electric dipole when placed in a uniform external field.

But they do experience a torque. As torque is the moment of force.

We define torque= as force* perpendicular distance between the forces.

When we try to rotate the electric dipole in an external field, it requires work done by some external force, and this work gets converted into the potential energy of the dipole of dipole moment P placed in an external uniform electric field.

Electrostatic of conductors

The conductor contains mobile charge carriers. In metallic conductors these charge carriers are electrons. The electrons from the valence shell of the atom of the metal behave like free electrons inside the metal and therefore conduct electricity. These free charge carriers form a kind of ‘gas’, they collide with each other and with the ions and move randomly in all directions. In the presence of an external field, they move and drift opposite to the direction of the field. The positive ions are made up of nuclei and bound electrons and remain held at fixed positions.

Properties regarding electrostatics of the conductors:

1. Inside the conductor, an electrostatic field is zero.

The conductor has free charges. In the presence of electric fields, these charges experience force and drift. But in the static condition, free charges are distributed in such a way that the Electric field inside the conductor is always zero.

2. At the surface of a charged conductor, the electrostatic field must be normal to the surface at every point.

If the electric field would not normal to the surface, the charges will move due to its tangential component and charges will no longer remain in static condition.

 

3. The interior of a conductor can have no excess charge in a static situation.

A conductor has an equal number of positive and negative charges inside it in any volume element. Any excess charge must reside on its surface and there is no net charge at any volume of it.

4. The electrostatic potential is constant throughout the volume of the conductor and has the same value (as inside) on its surface.

We have seen that E=0 inside the volume of the conductor, but does it mean that V should also be zero, certainly not!. As the derivative of constant is also zero. The conductor thus has a constant and same value of electrostatic potential inside and on the surface of it. And its value is different outside the conductor.

5. Electric field at the surface of a charged conductor  E=σ/ϵ0 and is normal to the surface.

If the conductor has charge density σ  on it. And we know that the electric field inside is zero. The electric field just outside the conductor will be normal to the surface and will have a value which is equal to σ/ϵ0. If σ>0, the direction of E will be outward normal and if σ<0, the direction of E will be inward normal.

6. Electrostatic shielding.

Consider a cavity of a conductor with no charge inside the cavity. The electric field inside the conductor is zero, whatever be the size or shape of the conductor or whatever be the charge on the conductor on the electric field outside the conductor.  This gives rise to a phenomenon called electrostatic shielding. If we place anything inside a conductor it will be shielded from any external field.

Dielectrics and polarization

Dielectrics are non conducting material, which has no charge carriers.

When we put a conductor in an external electric field, the charge inside it moves and arranges in such a way that the electric field induced inside it starts opposing the external field. This will happen till the internal induced field completely cancels the external field and the net field inside the conductor becomes zero.

But in the case of dielectric, in which free movement of charges is not possible  when put in an external field, the induced moment is  only due to stretching and re-orientation of the molecules of the dielectrics

Here also the induced field will try to oppose the external field but it cannot completely cancel it but can only reduce it.

Enet inside the dielectric = External field Eo - Induced field E.

Types of dielectric materials

 We have two kinds of dielectric materials, one is polar and the other is non-polar

In polar molecules, there is a dipole moment in individual molecules but in the absence of electricity, these molecules are randomly oriented in such a way that the net dipole moment is zero.

But when these molecules are placed in external E, these individual dipoles are aligned in the direction of the Electric field, basically re-orientation of the molecules happens and thus becomes polarized in the presence of the external electric field.

In nonpolar molecules, in the absence of an external electric field, there is no polarization. But when we apply an external electric field the molecule gets stretched and there happens a separation between positive and negative charges. So the polarization of the dipole happens.

The figure given below  it is shown how a dielectric is polarized in presence of an external electric field, note that all the charges shown are actually bound charges and not the free charges. The charges in the bulk of the dielectric neutralize each other and there is a net surface charge on the boundary of the dielectric, the surface gets a surface-bound charge of ±σb (  b represents bound charge here) . Thus creating an internal induced electric field within the dielectric material and therefore reducing the net field inside it.