Relation Between Electric Field and Electric Potential

In electrostatics, the concept of Electric field and electric potential plays an important role. Electric field or electric field intensity is the force experienced by a unit positive test charge and is denoted by E. Electric potential is the work done to move unit charge against the electric field or the electric potential difference is the work done by conservative forces to move a unit positive charge and is denoted by V.

Mathematically, the electric field and the potential is given by:

⇒\[E=\frac{F}{q}\]

⇒\[V=\frac{Kq}{r}\]

The relation between electric field and potential is similar to that of the relation between gravitational potential and the field. The relation between Electric field and Potential is generally given by -the electric field is the negative gradient of the electric potential.

Relation between Electric Field and Potential

The relation between electric field intensity and electric potential can be found with a small derivation given below. To establish the relation between Electric field and electric potential we will use basic concepts of electrostatics.

Derivation

To derive a relation between electric field and potential, consider two equipotential surfaces separated by a distance dx, let V be the potential on surface 1 and V-dV be the potential on surface 2. Let E be the electric field and the direction of the electric field is perpendicular to the equipotential surfaces.

(Image will be uploaded soon)

Let us consider a unit positive charge +1C near point B, the force experienced by the unit positive charge is given by:

⇒ F= qE ……(1)

Since q= +1C, equation (1) becomes,

⇒F = E………..(2)

Equation (2) is indicating that the magnitude and the direction of the electric field and the force are equal, which means the direction of force is also perpendicular to the equipotential surfaces.

If we move the charge from point B to point A, the work done in bringing the charge from point B to point A is given by,

 ⇒WBA = F.dx

 ⇒WBA = F dx Cosθ ……(3)

From equation (2) F = E, Substituting the value of F in equation (3) we get,

 ⇒WBA = E dx Cosθ …….(4)

Now, the force experienced is acting in the upward direction, but the displacement is in the downward direction, thus the angle between force and displacement is 180°. 

Therefore, the work done in bringing the point charge from point B to A is given by:

 ⇒WBA = ─ E dx……….(5)

From the definition of electric potential, we know that the electric potential is the work done in bringing a point charge from one point to another, thus we have:

⇒ WBA = VA- VB

Substituting the corresponding values of the potential at point A and B,

⇒WBA = V- (V-dV)=dV …………..(6)

Equating equation (5) and (6):

⇒ dV =  ─ E dx

⇒E=−dV/dx………………(7)

Therefore, the relationship between field and potential is the electric field due to a point charge is negative potential gradient due to the point charge. Equation (7) is known as the electric field and potential relation.

Equation (7) is the relation between electric field and potential difference in the differential form, the integral form is given by:

We have, change in electric potential over a small displacement dx is:

⇒ dV =  ─ E dx

⇒ \[\int dV =  -\int E.dx\]

⇒ΔV= VA-VB = \[- \int E.dx\]……..(8)

Equation (8) gives the integral form of a relation between the electric field and potential difference.

Case 1:

If the test charge is positive, then from the relationship between the electric field and electric potential, the potential gradient will be more near the charge.

Case 2:

If the test charge is negative, the potential gradient will be more as we move away from the test charge.

Case 3:

For an equipotential surface, the potential at every point on the surface will be the same, thus the potential gradient will be zero. The electric potential will be perpendicular to the electric field lines.

Examples:

1. Calculate the Electric Potential Due to a Point Charge at a Distance x From it.

Ans:

Given that, a point charge is placed at a distance x from point P(say). We are asked to calculate the potential at point P.

(Image will be uploaded soon)

We know that the electric field due to point charge is given by,

⇒\[E = \frac{kQ}{x^{2}}\]

From the relation between the electric field and the potential we have,

⇒ \[\int dV = - \int E.dx\]

 The limit of integration for LHS is 0 to V and for RHS infinity to x.

Substituting the value of E in the above expression,

⇒\[V^{v}0 = -\int (\frac{Kq}{x^{2}}).dx\]

⇒\[V= -Kq \int x^{2}.dx\]

On simplification we get,

⇒\[V = \frac{Kq}{x}\]

Therefore, by using the relation between the electric field and the potential it is convenient to derive results. The same relation can be derived by using the definition of electric potential.

2. The Electric Potential V at Any Point X, Y, and Z in Space is Given by V=3x2 Volts, then the Electric Field at Any Point (2,1,2) is?

Ans: 

Given, the potential at any point x,y, and z,

⇒ V=3x2

We have to find the electric field E at (2,1,2).

We know that,

⇒\[E=-\frac{dV}{dx}\]

⇒\[E=\frac{-d(3x^{2})}{dx}=6x\]

At x=2,

⇒ E=6(2)=12 V/m

Therefore, the electric field E at (2,1,2) is 12V/m.

Electricity Keeping us Alive

Everyone is familiar with electricity in modern times.

We get electricity from sockets in walls in our houses. It gives us a shock if we touch it.

Science has taught us that all matter is made up of very small particles called atoms. These atoms are made of two kinds of charge – positive and negative. The middle part of the atom contains the positive charge and flying around this is the negative charge. Most of the time the number of positive and negative charges in an atom are equal in number – they exist in pairs. When they are not equal in number, the extra negative charge leaves the atom and goes looking for its partner. These stray charges are electrons and are easier to move about. These moving electrons make up electricity.

Types of Electricity

Electricity is of two kinds – static and current. Static electricity, electrons are moved mechanically. In current electricity, the electrons move in a closed loop. When the loop is broken, electricity cannot flow.

Lightning is static electricity. During thunderstorms, a cloud can develop a buildup of negatively charged particles. Electrons repel each other – they are always looking for positively charged particles to pair up with. So, they try to get away from other electrons and make a leap to the biggest thing nearby – the earth. Lightening is a big spark, a group of electrons that jumped to the earth to find their positively charged mate. Lightning is the biggest spark that exists; a lightning bolt has over 20 million volts.

The electric circuit is like the blood circulation system in a body. Blood is pumped in the arteries by the heart and then comes back to the heart having traveled through arteries and veins. In an electric circuit, electric charges are the blood and the wires are the arteries and veins. Electric charges have a little amount of energy. It is measured in Volt. A handheld flashlight has 1.5 Volt. The wall socket at home has 120 Volts. When electrons move through this circuit, trying to move away from the negative charge towards the positive charge, this flow forms the current. In the flashlight, we use batteries with a plus sign on one side and a minus sign on the other side. The plus sign side of the battery is where the extra positive charges are present. The other side with the minus sign stores the extra negative charge – the free electrons. Given a path, they are ready to run to find their pair. When we press the button of a flashlight, the loop is completed, and the circuit becomes complete. The electrons find their path and race out of the battery to the positive charges. The bulb forms a part of their path – the circuit. On their way, they make the wire inside the bulb very hot and it glows.

Similarly, when we touch the wall socket, lots of electrons (the electric current) flow through our body and this is why we get the shock.

Uses of Electricity in Daily Life

• The electrical field is used to push electrons through wires. This electricity powers the fans in our houses, the air conditioner, the computer, the charging sockets, and even the internet. 

• Electromagnetic waves use electric fields. This allows us to receive radio signals in cars and houses. This is also how we communicate via satellites, get weather information. 

• Since light is also an electromagnetic wave, the electric field helps us see in darkness. Everyday machines use electrical fields like motors and generators and the television. Without electrical fields, we would have no electricity. 

• Cars and airplanes use alternators and magnetos inside them that work through electricity. 

• The electric field also shields us from cosmic radiation and practically, we would all be dead without electric fields and electricity.