Electric Potential

Dr. Nur Islami bin Abd. Rahman

Electrostatics And Potential Energy

The electrostatic force is a conservative force.

This means that the work it does on a particle depends only on the initial and final position of the particle, and not on the path followed.

The introduction of the potential energy is useful since it allows us to apply conservation of mechanical energy which simplifies the solution of a large number of problems.

Electrostatics And Potential Energy

Gravitational force (mog) acts on a free falling mo

Electrostatic force (qoE) acts on a free falling positive qo

Newtons law and coulombs law are identical

Difference in its gravitational potential energy (U) :

We can also use the equation above as a definition of the difference in the electric potential energy of a test charge q0

The difference in the electric potential energy of a test charge between two points is independent of the path taken between those points. So, we can see here that electrostatic force is a conservative force and gravitational force as well.

If we choose to locate point i at a very far (large distance) from the charge, and we assign the value zero to the Ui = 0, and Uf = U so:

The potential energy U of a test charge qo at any point is equal to the negative of the work W done on the charge by the electric field.

Sample problem

Gas balloon of helium with charge of q = - 5.5 x 10-8 C, rises vertically into the air by distance d = 250 m from initial position to the final position. If the natural electric field on the earth surface is E = 150 N/C and is directed downward, what is the difference in electric potential energy of the balloon between position i and f? (Uf Ui = - W = - 4.3 x 10

-3 J)

The Electric Potential

The potential energy per unit charge at a point in an electric field is called electric potential V at the point.

The potential difference V between any two points i and f in an electric filed is:

The Electric Potential

The work Wif done by the electric field on the positive test charge as the charge moves from point i to f may be positive, negative or zero when the electric potential at f is higher, lower or the same as electric potential at i, respectively.

Finally the potential at any point in an electric field is:

The Electric Potential

The electric field around a charged rod can be described not only by an (vector) electric field E but also by a scalar quantity, the electric potential V.

The SI unit of potential difference is joule/coulomb equal to volt ( 1 volt = 1 joule/coulomb)

Equipotential

Point in space that all have the same potential is called an equipotential surface

Equipotential

Point in space that all have the same potential is called an equipotential surface

Calculating The Potential From the Field

We can calculate the potential difference between any two points i and f in an electric field if we know the field vector E at all positions along any path connecting those points.

Consider an arbitrary electric field and a test charge qo that moves along the path from point i to point f. at any point, an electrostatic force qoE acts on the charge as it moves through a displacement ds.

Calculating The Potential From the Field

Because the electric force is conservative, all path will provide the same result. And if we chose the potential Vi at point i to be zero, then we get

Sample problem

Figure below shows two points i and f with distance d in a uniform electric field E. Find the potential difference Vf Vi by moving a positive test charge qo from i to f

Potential due to a Point Charge

Figure below shows an isolated positive point charge of magnitude q. We wish to find the potential V at point P due to q, radial distance r from that charge. This potential is to be determined relative to a zero potential that we assign to infinity.

Potential due to a Point Charge

The test charge qo moves from infinity to point P. We make it simple path due to the path followed by the charge is not matter. The angel between E and ds will be 180, ds will equal to -dr

Potential due to a Group of point Charge

We can find the net potential due to a group of point charges at any given point with the help of the superposition principle. For n charges, the net potential is:

Here qi is the value of the ith charge and ri is the radial distance of the given point from the ith charge. The sum in equation 9 is an algebraic sum not a vector sum.

Sample problem

What is the potential at point P, located at the center of the square of point charges below: (352V)

Potential due a continuous charge distribution

When a charge distribution is continuous, we must choose a typical differential element of charge dq, determine the potential dV at P due to dq, and then integrate over the entire continuous charge distribution.

Potential due a continuous charge distribution

Line of charge

Potential due a continuous charge distribution

Line of charge

Potential due a continuous charge distribution

Sphere of charge

Radius = r

Charge = Q

Calculate V,

To make it simple, E must be obtained using Gausss law

Calculating the Field from the Potential

Embark from this equation:

We can get field:

ds can be applied to other direction such as dx, dy and dz

Sample problem

The potential at any point on the axis of a charged disk is given by

Derive an equation for the electric field at any point on the axis of the disk

The Van de Graaff accelerator

The Van de Graaff accelerator is an arrangement for generating potential differences of the order of several million volts.

By allowing charged particles such as electrons or protons to fall through this potential difference, a beam of energic particles ca be produced.

The Van de Graaff accelerator

A Van de Graaff generator is an electrostatic generator which uses a moving belt to accumulate very high voltages on a hollow metal globe on the top of the stand.

It was invented by American physicist Robert J. Van de Graaff in 1929. The potential difference achieved in modern Van de Graaff generators can reach 5 megavolts.

The Van de Graaff accelerator

1. hollow metallic sphere (with positive charges)

2. electrode connected to the sphere, a brush ensures contact between the electrode and the belt

3. upper roller (Plexiglass) 4. side of the belt with positive

charges 5. opposite side of the belt with

negative charges 6. lower roller (metal) 7. lower electrode (ground) 8. spherical device with negative

charges, used to discharge the main sphere

9. spark produced by the difference of potentials

The Van de Graaff accelerator

Review & Summary

The change U in the electric potential energy U of a point charge as the charge moves from an initial point i to a final point f in electric field is

where the work Wif is that done by the electric field on the point charge.

If the zero potential energy is defined to be at infinity, the electric potential energy U of the point charge at a particular point is

Wf is the work done by the electric field on the point charge as the charge moves from infinity to the particular point

Review & Summary

The potential difference V between any two points i and f in an electric filed is:

The potential (volt=J/C) at any point in an electric field is:

The point on an equipotential surface all have the same potential

Finding V from E

Review & Summary

Potential due to point charge

Potential due to a Group of point Charge

Potential due a continuous charge distribution

Calculating the Field from the Potential