Electric Potential Dipole

Introduction

When we study a charge or a system of charge, we see that every charge has a definite amount of electric field which exerts the electrostatic force of repulsion or attraction on other charges and charged particles according to their nature. The effect of electricity can be experienced with the help of a unit charge, also known as a test charge. The electric field around a charge or system of charge can be described in two forms; The electric field and The electric potential.

What is the Electric Potential of a Dipole and System of Charges?

The electric potential of a dipole is the electrostatic potential that arises due to a dipole. Now, first of all, we should clarify what is electric potential? In simple words, we can say that the electric potential for a point in an electric field is the piece of work we have to do for moving a point positive charge of magnitude one coulomb from infinity position to that considered point in which the effect of the electric field is present which continues exerting electrostatic force. Hence, electric potential can be expressed as work done on a unit charge.

The standard measurement unit of Electric potential is Volt. Hence, one-volt electric potential point is the one joule of work that has been done to bring a unit positive charge from an infinite far position to a target point under the influence of force due to an electric field.

V=WqV=Wq

The electric dipole is a system of two charges having the same value and opposite nature and these are connected by a line that passes through their center. The length of the line that passes through its center is called dipole length.

To measure the strength of electric dipole we use the term dipole moment. A Dipole moment can be defined as a vector whose magnitude is the product of charges and the total separation between them and the direction of the vector will be along the axis from negative to positive. It is denoted by p.

The SI unit of dipole moment is coulomb metre.

On the other hand, if we use a system of charges having more than 2 charges then the electric potential is called the electric potential developed by the system of charges.

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Drive an expression for electric potential at a point due to electric dipole

The electric potential created at any point, such as P because of an electric dipole can be expressed for two points; axial point and equatorial point.

• Electric Potential developed by a dipole at an axial point

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In the above diagram, there is a dipole having pair of charges; positive at A and negative at B, separated by some distance denoted by d which is equal to 2a. There is point P which lies on the axis of the dipole which is r distance away from the center of the dipole.

Now, the electric potential for point P by both charges is

V=V1+V2V=V1+V2

V1=14πϵ0.−qAPV1=14πϵ0.−qAP

Now,

V2=14πϵ0.+qBPV2=14πϵ0.+qBP

Total Potential at P,V=V1+V2V=V1+V2

So,

V=14πϵ0.−qAP+14πϵ0.+qBPV=14πϵ0.−qAP+14πϵ0.+qBP

We know that AP=r+d2AP=r+d2 and BP=r−d2BP=r−d2 whereas d2=ad2=a.

Thus,V=14πϵ0.−qr+a+14πϵ0.+qr−aThus,V=14πϵ0.−qr+a+14πϵ0.+qr−a

V=q4πϵ0.{1r−a−1r+a}V=q4πϵ0.{1r−a−1r+a}

V=q4πϵ0.{(r+a)−(r−a)r2−a2}V=q4πϵ0.{(r+a)−(r−a)r2−a2}

V=q4πϵ0.{q×2ar2−a2}V=q4πϵ0.{q×2ar2−a2}

However, p=q×2ap=q×2a

Hence,V=q4πϵ0.{pr2−a2}V=q4πϵ0.{pr2−a2}

If the dipole is very small then a2<<r2a2<<r2

So,

V=q4πϵ0.pr2V=q4πϵ0.pr2

• Electric Potential at an equatorial point of the dipole

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Let a dipole of a pair of charges having opposite nature and equal magnitude, which is -q and +q, both charges are 2l distant from each other. There is a random point P which lies position linear to the perpendicular bisector of the dipole and at a distance of P is r from the center of the dipole.

Now, the distance of point P from -q and +q is the same which is r2+l2−−−−−√r2+l2

According to the formula of the electric potential created at position P because of both charges is

V=V1+V2V=V1+V2

V=14πϵ0.−qAP+14πϵ0.+qBPV=14πϵ0.−qAP+14πϵ0.+qBP

Here, both AP and BP are the same.

So,V=14πϵ0.−qr2+l2−−−−−√+14πϵ0.+qr2+l2−−−−−√So,V=14πϵ0.−qr2+l2+14πϵ0.+qr2+l2

Thus,−14πϵ0.qr2+l2−−−−−√+14πϵ0.qr2+l2−−−−−√=0Thus,−14πϵ0.qr2+l2+14πϵ0.qr2+l2=0

V=0V=0

Therefore, from the above expression find out that the electric potential at the equatorial point of an electric dipole is zero.