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What is Electrostatic Potential – Definition and Explanation
When two positively charged insulated conductors are connected by a metallic wire, charge flows from one conductor to the other. The direction of flow of charge does not depend on the amount of charge on them, but depends on a specific electric condition of the two conductors. This electric property of a charged body is called electric potential. So electric potential of a body is the electric property which determines whether charge will flow from this body to any other body connected to it or from any other body to itself,
i) Analogy between electric potential and hydrostatic level or height of water column: Two interconnected vessels A and B of different sizes contain water of different amounts held at the different heights [Fig.]. When connection is established by opening the stop-cock D, water flows from the vessel with higher level of water to the other, until the levels are equal. So flow of water does not depend on the quantity of water, but depends on the height of water column or pressure of water. Similarly flow of charge or electricity between two bodies depends on the difference of their potential but not on the quantity of charge they have.
Water flows from higher to lower level, i.e., from higher to lower pressure. Similarly electric charge flows from a body at higher potential to another at lower potential. So electric potential may be compared with height of water as well as the pressure of water and amount of charge with the quantity of water.
ii) Analogy between electric potential and temperature: Heat flows from a hotter body to a colder one in contact till their temperatures are equalised. This occurs even if the heat content of the colder body is more than that of the hotter one. Thus flow of heat depends only on the temperature difference of the bodies in contact, but not on their heat contents. In this context, electric potential may be compared with temperature and amount of charge with heat.
It may be noted that while flow of water or heat is unidirectional, flow of charge depends on its nature. According to the sign convention, positive charge flows from higher to lower potential and negative charge from lower to higher potential. No charge however flows from one body to another if they are at the same potential.
Potential at a Point in an Electric Field
Consider an isolated charge in a region where no electric field is present. Now, due to the presence of the charge, an electric field develops in that region. When a second charge is brought in to that region, an electric force acts on it. To displace the second charge within this region, work has to be done, either by some external agent or by the field itself, depending on the nature of the charge.
Hence it can be concluded that a medium containing a charge acquires some property for which some work has to be done to displace another charge in that medium. This property is known as electric potential.
To define electric potential we need an infinitesimal test charge (q) that does not disturb the priorly existing charge (Q) which causes an electric field. In this context, we shall name this test charge as a unit charge, i.e., q = 1. Also we assume though there is no boundary of the electric field, the potential beyond the field i.e., at infinity is zero. The initial position of the test charge is assumed to be at infinity which is its reference point.
From this reference point if a unit positive charge is to be taken in an electric field one has to apply a minimum force (just equal but opposite to that electric field force) to take the test charge from infinity to a specific point in the electric field and hence a work is to be done. As in such work done no net force is applied on the test charge it has no acceleration.
Definition: The potential at any point in an electric field is defined as the work done by an external force in bringing a unit positive charge without acceleration from infinity to that point.
Suppose, potential at any point in an electric field is V.
By definition, work done to bring a unit positive charge from infinity to that point = V.
Therefore, work done to bring a charge q from infinity to that point, W = V ᐧ q ….. (1)
work done = potential × charge
This work done is stored up as electric potential energy of the system consisting of the charge and the external electric field. So, electric potential energy = potential x charge
i. e., electric potential energy of a unit charge placed at a point in an electric field is the electric potential at that point.
In equation (1), since work done and charge are both scalar quantities, so, potential as well as potential energy are scalar quantities.
The potential of a positively charged body is said to be positive, because work has to be done by some external agency to bring a unit positive charge from infinity to any point in the electric field. This is to overcome the force of repulsion between the positively charged body and the unit positive charge (test charge). So energy is stored in the system consisting of the positively charged body and test charge. Therefore both W and V are positive.
On the other hand, the potential of a negatively charged body is said to be negative, because a unit positive charge moves itself closer to the charged body due to attraction, i.e., work is done by the attractive force. So both W and V are negative.
Potential difference between two points: The potential difference between two points in an electric field is defined as the amount of work done by an external force to bring a unit positive charge without acceleration from one point to the other.
Let VA and VB be the electric potentials at the points A and B , respectively in an electric field. If WAB be the work done in bringing the charge q from A to B, the potential difference between the two points is given by,
VB – VA = \(\frac{W_{A B}}{q}\)
Obviously, if WAB is positive, VB > VA.
If WAB is negative, VB < VA
If WAB is zero, VB = VA.
Potential difference between two points is independent of the path connecting the points:
Potential difference between two points in an electric field does not depend on the path followed from one point to another. The path may be straight or curved but the amount of work done, i.e., potential difference remains the same.
Proof:
Supposed and B are two points in an electric field [Fig.] and an external agent performs an amount of work W1 to bring a unit positive charge from the point A to the point B along the path A CB. Now this unit positive charge is brought back to A from B along the path BDA. In this case, let us suppose that the work done by the electrical force = W2, where W2 ≠ W1 . Let W2 > W1.
So the amount of work done in the whole process is (W2 – W1); i.e., this net amount of energy will be surplus. If the unit positive charge is taken repeatedly along the closed path ACBDA, energy would be evolved continuously. But according to the principle of conservation of energy, this is not possible. So (W2 – W1) should be equal to zero, i.e., W2 = W1.
So in an electrostatic field
1. total work done in moving a charge around a closed path is zero and
2. the potential difference between two points is independent of the path along which a charge may be brought from one point to the other So, like gravitational force, electrical force is also conservative in nature.