General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor, M.Sc., Ph.D.

Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294

Electric Field


In this chapters we will learn

concept of  “electric field”

To calculate field generated by a point charge.

Application of principle of superposition to determine the electric field created by a collection of point charges as well as continuous charge distributions.

Calculate electric force from electric field

Notion of “electric dipole.”

To determine net force and net torque, exerted on an electric dipole by a uniform electric field.

What is a Field?

Electric Field_1.gif

Assignment of a quantity or number to each point in space is called the field of that quantity.

What is Electric Field?

Consider a ball hanging with a thread.

Put some positive charge on it (Click the PLAY button to charge the ball), What we observe?

The ball gets pulled or pushed towards right by some invisible force.

Now increase the charge by "Change charge" slider. What happens now?

What can we conclude?

It seems that there is "something invisible" at the location of ball, waiting to interact with the charge of a particle .
This "something invisible" is called the "Electric Field."

We know that this "something invisible" is produced by other charges in space, therefore we can state:

"Electric Field" is a kind of invisible influence of a charge or charges at different locations in space, which can be experienced by other charge or charges.

Definition of Electric Field

"Electric field" is invisible but we can still measure it.

Let us release a known positive test charge Electric Field_3.gif of mass m at a point P.

It moves, therefore we can measure its acceleration Electric Field_5.gif.

From acceleration Electric Field_6.gif, we can compute the force Electric Field_7.gif experienced by test charge Electric Field_8.gif due to invisible "electric field" due to other surrounding charged objects.

Electric Field_9.gif

Now the Electric field vector Electric Field_10.gif at releasing point P is defined as

Electric Field_11.gif

SI units of electric field are N/C (Newton/Coulomb)

Test charge can be positive or negative.

By convention, the direction of Electric Field_12.gif points in the direction of measured force Electric Field_13.gif experienced by a positive test charge Electric Field_14.gif.

Direction of Electric Field_15.gif will points opposite to measured Electric Field_16.gif if test charge Electric Field_17.gif is negative.

Checkpoint 1 (Electric Field)

At an observation point we found that direction of electric field Electric Field_18.gif points to North.
(a) What will be the direction of force Electric Field_19.gif experienced by a positive charge at that point.
(b) What will be the direction of force Electric Field_20.gif experienced by a negative charge at that point.

Electric field of a point charge

To compute electric field at point P due to a charge Q, first we need to calculate force Electric Field_21.gif on a test charge Electric Field_22.gif.

Electric Field_23.gif

As per Coulombs law, force Electric Field_24.gif on Electric Field_25.gif due to Q at point P is given as

Electric Field_26.gif

Now electric field is defined as

Electric Field_27.gif

Therefore electric field of a point charge Q at point P is

Electric Field_28.gif

If Q is a positive charge, direction of electric field Electric Field_29.gif is away from the charge Q.

If Q is a negative charge, direction of electric field Electric Field_30.gif is towards  the charge Q.

Electric Field Lines

Electric field can be represented by drawing field lines.

Electric Field_31.gif Electric Field_32.gif

Electric field lines represent the path of a positive test charge (generally test charge amount is very small).

Electric field lines extend away from positive charge (where they originate) and towards negative charge (where they terminate).

These days we no longer attach much reality to these lines.

But these electric field of lines still provide a nice visualization of electric field.

Principle of superposition for the electric field

To measure electric field at any point due to many charges, we need to calculate net electrostatic force on a test charge Electric Field_33.gif.

Electrostatic force on any charge Electric Field_34.gif due to other charges in the surroundings follows the principle of superposition. (Alt click and create more surrounding charges)

Net electrostatic force Electric Field_36.gif on a point test charge Electric Field_37.gif due to other charges in the surrounding is given as

Electric Field_38.gif

Electric field Electric Field_39.gif at the position of test charge Electric Field_40.gif due to Electric Field_41.gif, Electric Field_42.gif, Electric Field_43.gif, etc. is defined as

Electric Field_44.gif

It shows that electric field, at the location of test charge Electric Field_45.gif, is again obtained by superposition of electric field due to surrounding charges Electric Field_46.gif, Electric Field_47.gif,  Electric Field_48.gif, etc.

Electric Field_49.gif

If there are N number of charges, electric field at any point P will be vector sum of fields at that point due to all the charges.

Electric Field_50.gif

Checkpoint 2 (Electric Field due to a point charge)

The figure here shows a proton p and an electron e on an x axis.
What is the direction of the electric field due to the electron at
(a) point S?
(b) point R?
What is the direction of the net electric field at
(c) point R?
(d) point S?

Electric Field_51.gif

Electric Dipole

A system of two equal charges of opposite sign (±q) separated by distance s apart is known as an "electric dipole."

Electric Field_52.gif

For every electric dipole we associate a vector Electric Field_53.gif known as the electric “dipole moment"

Electric Field_54.gif is defined as

Electric Field_55.gif

The direction of Electric Field_56.gif is along the line that connects the two charges and points from q to +q.

Electric Field of a dipole :

Field lines start from the positive charge and ends at the negative charge.

Electric Field_57.gif

Naturally occurring Electric Dipoles

Many molecules have a built-in electric dipole moment. Water molecule Electric Field_58.gif and HCl are two such examples.

Water Molecule Electric Field_59.gif

Electric Field_60.gif involves ten shared electrons (8 from Oxygen and 2 from two Hydrogen atoms).

Eight protons in Oxygen atom try to attract these electrons more towards Oxygen atom.

Electric Field_61.gif

This shifting of 10 electrons more towards Oxygen, makes the Oxygen atom little negative and the two H atoms little positive.

HCl Molecule

HCl involves eighteen shared electrons (17 from Chlorine and 1 from Hydrogen atom).

Seventeen protons in Chlorine atom try to attract one Hydrogen electron more towards Chlorine atom.

Electric Field_62.gif

This shifting of 18 electrons more towards Chlorine, makes the Cl atom little negative and the H atom little positive.

Yellow arrow is the dipole moment vector Electric Field_63.gif.

Electric Field of an Electric Dipole

Consider a dipole with equal and opposite charges (±q) separated by distance s.

Electric Field_64.gif

Net electric field Electric Field_65.gif at point P along z-axis should be sum of electric fields due to the two charges at that point.

Electric Field_66.gif

Let us compute Electric Field_67.gif due to positive charge Electric Field_68.gif.

Electric Field_69.gif

Electric Field_70.gif

Electric Field_71.gif

Electric Field_72.gif

Similarly electric field Electric Field_73.gif due to negative charge Electric Field_74.gif can be computed.

Electric Field_75.gif

Electric Field_76.gif

Electric Field_77.gif

Electric Field_78.gif

Net electric field Electric Field_79.gif at P can be computed as

Electric Field_80.gif

Magnitude of electric field Electric Field_81.gif at P is

Electric Field_82.gif

Electric Field_83.gif

After simple algebra we get

Electric Field_84.gif

If s<<x, we can neglect Electric Field_85.gif in comparison to x, therefore the field at distance much larger than separation of dipole we get

Electric Field_86.gif

In terms of dipole moment p=q s, we can write

Electric Field_87.gif

Electric field due to a Continuous Charge Distribution

Divide the charge distribution into small elements.

Electric Field_88.gif

Treat each element as a point charge.

Calculate electric field Electric Field_89.gif due to each element at point P.

Electric Field_90.gif

Vectorially add the fields due to all the elements.

Electric Field_91.gif

If the charge distribution is uniform with volume charge density ρ = Q/V, Q is the total charge and V is the total volume. Above integral can be written as

Electric Field_92.gif

In three dimensions, we need to compute three integrals, one for each component.

Checkpoint 3 (Electric Field due to a line charge)

The figure here shows three nonconducting rods, one circular and two straight. Each has a uniform charge of magnitude Q along its top half and another along its bottom half.
For each rod, what is the direction of the net electric field at point P?

Electric Field_93.gif

Electric field of a uniformly charged ring

To compute electric field at a point P on the axis passing through the center of a uniformly charged ring of radius R, we can divide the ring in equal segments of length ds.

Let us assume that λ is the linear charge density of the ring.

Electric Field_94.gif

The electric field Electric Field_95.gif generated by element ds has two components, one parallel to z-axis and other perpendicular to z-axis.

Due to the presence of identical element on diagonally opposite side, the perpendicular components get canceled.

Total electric field is due to the addition of z-components of each element.

Electric Field_96.gif

Electric Field_97.gif

Electric Field_98.gif

Electric Field_99.gif

Electric Field of a Charged Disk

Consider a uniformly charged disk of radius R with surface charge density σ.

Electric Field_100.gif

Since it is a disk so let us consider its surface charge density Electric Field_101.gif.

We have already calculated the electric field at point P due to a ring.

Let us consider a ring of width dr, radius r. Total charge on the ring of area dA will be

Electric Field_102.gif

The electric field dE generated by this ring along z-axis is

Electric Field_103.gif

Total electric field is computed by integration

Electric Field_104.gif

Electric Field of an infinite sheet

Electric field of a uniformly charged disk with σ surface charge density is given as    

Electric Field_105.gif

What if the radius R? The disk becomes an infinite sheet.

Electric field for such a sheet will be

Electric Field_106.gif

Similarly for a finite sheet if we want to compute electric field very close to the sheet then we can assume z<<R.

In such situation we again get      

Electric Field_107.gif

Electric Field_108.gif

Is Electric Field Real?

Simple reason for us to introduce the concept of Electric field was:

Once we know the electric field Electric Field_109.gif at some location, we know the electric force Electric Field_110.gif acting on any charge q that we place there.

Electric field is a kind of "invisible influence" of a charge or charges at different locations in space, which can be experienced by other charge or charges.

However, Einstein's special theory of relativity predicts that nothing can move faster than the speed of light Electric Field_111.gif/s), therefore influence of a charge (Electric field) too can not move faster than this speed.

With the assumption of finite speed of electric field let us watch its behavior with time in following simulations.

Create a charge at a location and watch how the "invisible influence" (electric field) travels with time.

Now destroy this charge and see how the electric field gets wiped out with time?

You can see that even after the destruction of charge, its electric field exists for finite time at various locations of space.

This shows that Electric Field concept is necessary to explain this phenomenon.

Electric Field of a charge exists even after destruction of the charge.

Electric Dipole in a Uniform Electric Field

Net Force

Electric Field_113.gif

Consider an electric dipole in uniform electric field Electric Field_114.gif. Let us see if it is in equilibrium or not?
First we calculate net force Electric Field_115.gif on the dipole.
Force on the negative charge is Electric Field_116.gif and is given as

Electric Field_117.gif

Similarly force on the positive charge will be Electric Field_118.gif

Electric Field_119.gif

Electric Field_120.gif

This shows net force on the dipole is zero.

Net Torque

Electric Field_121.gif

Now let us calculate net torque Electric Field_122.gif on the dipole First calculate torque Electric Field_123.gif on the negative charge.

Electric Field_124.gif

Minus sign indicates that rotation is clockwise.
Now let us calculate torque Electric Field_125.gif on the positive charge.

Electric Field_126.gif

Electric Field_127.gif again produce rotation in clockwise direction.
Net torque Electric Field_128.gif is

Electric Field_129.gif

Electric Field_130.gif

We can write net torque acting on the dipole in terms of dipole moment

Electric Field_131.gif

In vector Notation we can write

Electric Field_132.gif

Electric Field_133.gif

Electric dipole in a uniform field can not move but it can rotate.

Behavior of a water molecule in electric field.

Click PLAY to put the electric field on. PAUSE and then use RESET button to put it off.

Potential Energy of an Electric Dipole in a Uniform Electric Field

Suppose an electric dipole rotate by an angle θ in an electric field Electric Field_135.gif. The change in potential energy ΔU of the dipole will be  

Electric Field_136.gif

Electric Field_137.gif

Electric Field_138.gif

Electric Field_139.gif

Stable equilibrium point of a dipole :

When a dipole makes 0° angle with electric field, its potential energy is minimum.

Electric Field_140.gif

Variation of dipole potential energy with angle θ.

For a dipole in electric field 0° is a stable equilibrium point

Unstable equilibrium point of a dipole :

When a dipole makes 180° angle with electric field, its potential energy is maximum.

Electric Field_141.gif

Variation of dipole potential energy with angle θ.

For a dipole in electric field 180° is an unstable equilibrium point

Simulation of a dipole in a uniform electric field

Change initial angle θ between dipole moment Electric Field_142.gif and electric field Electric Field_143.gif to zero and apply electric field (Click PLAY).

Watch the dipole, it is in its stable equilibrium position (Minimum energy position) so nothing happens.

Now change the angle to a different value and watch the motion (Click PAUSE, RESET and PLAY again).

Change initial angle θ between dipole moment Electric Field_145.gif and electric field Electric Field_146.gif to 180° and apply electric field (Click PAUSE, RESET and PLAY again).

Watch the dipole in its unstable equilibrium position (Maximum energy position) , again nothing happens.

Now change the angle by small value and watch the motion (Click PAUSE, RESET and PLAY again).

Dipole returns back to its stable equilibrium position.

Checkpoint 4 (Electric Dipole)

The figure shows four orientations of an electric dipole in an external electric field. Rank the orientations according to
(a) the magnitude of the torque on the dipole and
(b) the potential energy of the dipole, greatest first.

Electric Field_147.gif