In the series of NCERT study material we are introducing CBSE 12th Electrostatic Potential and Potential Difference 1. You may also visit our dedicated page for 12th physics for other chapter notes and worksheet along with sample papers.
Electrostatic Potential and Potential Difference 2
The electric field around a charge can be described in two wavs :
(i) by electric field (E), and E
(ii) by electrostatic or electric potential (V).
The electric field E is a vector quantity, while electric potential is a scalar quantity. Both of these quantities are the characteristic properties of any point in a field and are inter-related.
1.Develop the concepts of potential difference and electric potential. State and define their SI units.
Potential difference. As shown in Fig. 2.1, consider a point charge + q located at a point O. Let A and B be two points in its electric field. When a test charge q0 is moved from A to B, a work WAB has to be done in moving against the repulsive force exerted by the
charge + q. We then calculate the potential difference between points A and B by the equation :
The potential difference between two points in an electric field may be defined as the amount of work done in moving a unit positive charge from one point to the other against the electrostatic forces.
In the above definition, we have assumed that the test charge is so small that it does not disturb the distribution of the source charge. Secondly, we just apply so much external force on the test charge that it just balances the repulsive electric force on it and hence does not produce any acceleration in it.
Unit of potential difference is volt (V). It has been named after the Italian scientist Alessandro Volt.
or 1 V = 1 Nm C-1 = 1 JC-1
Hence the potential difference between two points in an electric field is said to be 1 volt if l joule of work has to be done in moving a positive charge ofl coulomb from one point to the other against the electrostatic forces.
Electric potential. The electric potential at a point located far away from a charge is taken to be zero.
In Fig. 2.1, if the point A lies at infinity, then VA = 0, so that
where W is the amount of work done in moving the test charge q0 from infinity to the point B and VB refers to the potential at point B.
So the electric potential at a point in an electric field is the amount of work done in moving a unit positive charge from infinity to that point against the electrostatic forces.
Electric potential = Workdone/ Charge
SI unit of electric potential is volt (V). The electric potential at a point in an electric field is said to be 1 volt if one joule of work has to be done in moving a positive charge of 1 coulomb from infinity to that point against the electrostatic forces.
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12th Electrostatic Potential and Potential Difference 1
ELECTRIC POTENTIAL DUE TO A POINT CHARGE
Electric potential due to a point charge. Consider a positive point charge q placed at the origin O. We wish to calculate its electric potential at a point P at distance r from it, as shown in Fig. 2.2. By definition, the electric potential at point P will be equal to the amount of work done in bringing a unit positive charge from infinity to the point P.
Fig. 2.2 Electric potential due to a point charge.
Suppose a test charge q0 is placed at point A at distance x from O. By Coulomb’s law, the electrostatic force acting on charge q0 is
The force acts away from the charge q. The small work done in moving the test charge q0 from A to B through small displacement against the electrostatic force is
Hence the work done in moving a unit test charge from infinity to the point P, or the electric potential at point P is
Clearly, V ∝ 1 / r. Thus the electric potential due to a point charge is spherically symmetric as it depends only on the distance of the observation point from the charge and not on the direction of that point with respect to the point charge. Moreover, we note that the potential at infinity is zero.
Figure 2.3 shows the variation of electrostatic potential (V ∝ 1 / r) and the electrostatic field (E ∝ 1 / r2) with distance r from a charge q.
Fig. 2.3 Variation of potential V and field E with r from a point charge q.
12th Electrostatic Potential and Potential Difference 1
ELECTRIC POTENTIAL DUE TO A DIPOLE
Electric potential at an axial point of a dipole. As shown in Fig. 2.4, consider an electric dipole consisting of two point charges – q and + q and separated by distance 2 a. Let P be a point on the axis of the dipole at a distance r from its centre O.
Fig. 2.4 Potential at an axial point of a dipole.
Electric potential at point P due to the dipole is
Electric potential at an equatorial point of a dipole. As shown in Fig. 2.5, consider an electric dipole consisting of charges – q and + q and separated by distance 2 a. Let P be a point on the perpendicular bisector of the dipole at distance r from its centre O.
Fig. 2.5 Potential at an equatorial point of a dipole.
Electric potential at point P due to the dipole is
Electric potential at any general point due to a dipole. Consider an electric dipole consisting of two point charges – q and + q and separated by distance 2 a, as shown in Fig. 2.6. We wish to determine the potential at a point P at a distance r from the centre O, the direction OP making an angle θ with dipole moment .
Let AP = r1 and BP = r2.
Net potential at point P due to the dipole is
12th Electrostatic Potential and Potential Difference 1
Differences between electric potentials of a dipole and a single charge.
1. The potential due to a dipole depends not only on distance r but also on the angle between the position vector of the observation point and the dipole moment vector . The potential due to a single charge depends only on r.
2. The potential due to a dipole is cylindrically symmetric about the dipole axis. If we rotate the observation point P about the dipole axis (keeping r and θ fixed), the potential V does not change. The potential due to a single charge is spherically symmetric.
3. At large distance, the dipole potential falls off as 1 / r2 while the potential due to a single charge falls off as 1 / r.
E the electric potential at a point whose position vector is , would be
Electric potential is a scalar quantity while potential gradient is a vector quantity.
The electric potential near an isolated positive charge is positive because work has to be done by an external agent to push a positive charge in, from infinity.
The electric potential near an isolated negative charge is negative because the positive test charge is attracted by the negative charge.
The electric potential due to a charge q at its own location is not defined – it is infinite.
Because of arbitrary choice of the reference point, the electric potential at a point is arbitrary to within an additive constant. But it is immaterial because it is the potential difference between two points which is physically significant.
For defining electric potential at any point, generally a point far away from the source charges is taken as the reference point. Such a point is assumed to be at infinity.
As the electrostatic force is a conservative force, so the work done in moving a unit positive charge from one point to another or the potential difference between two points does not depend on the path along which the test charge is moved.
12th Electrostatic Potential and Potential Difference 1
Electric Potential
1. Potential difference = or V =
2. Electric potential due to a point charge q at distance r from it,
3. Electric potential at a point due to N point charges,
4. Electric potential at a point due to a dipole,
Units Used
Charge q is in coulomb, distance r in metre, work done W in joule and potential difference V in volt.
The quantity is the rate of change of potential with distance and is called potential gradient. Thus the electric field at any point is equal to the negative of the potential gradient at that point. The negative sign shows that the direction of the electric field is in the direction of decreasing potential. Moreover, the field is in the direction where this decrease is steepest.
Properties relating electric field to electric potential :
(i) Electric field is in that direction in which the potential decrease is steepest.
(ii) The magnitude of electric field is equal to the change in the magnitude of potential per unit displacement (called potential gradient) normal to the equipotential surface at the given point.
SI emits of electric field. Electric field at any point is equal to the negative of the potential gradient. It suggests that the SI unit of electric field is volt per metre. But electric field is also defined as the force experienced by a unit positive charge, so SI unit of electric field is newton per coulomb.
12th Electrostatic Potential and Potential Difference 1
EQUIPOTENTIAL SURFACES AND THEIR PROPERTIES
Equipotential surface. Any surface that has same electric potential at every point on it is called an equipotential surface. The surface may be surface of a body or a surface in space. For example, as we shall see later on, the surface of a charged conductor is an equipotential surface. By joining points of constant potential, we can draw equipotential surfaces throughout the region in which an electric field exists.
Properties of equipotential surfaces :
1. No work is done in moving a test charge over an equipotential surface.
2. Electric field is always normal to the equipotential surface at every point. If the field were not normal to the equipotential surface, it would have a non-zero component along the surface. So to move a test charge against this component, a work would have to be done. But there is no potential difference between any two points on an equipotential surface and consequently no work is required to move a test charge on the surface. Hence the electric field must be normal to the equipotential surface at every point.
3. Equipotential surfaces are closer together in the regions of strong field and farther apart in the regions of weak field. We know that electric field at any point is equal to the negative of potential gradient at that point.
4. No two equipotential surfaces can intersect each other. If they interesect, then there will be two values of electric potential at the point of intersection, which is impossible.
EQUIPOTENTIAL SURFACES OF VARIOUS CHARGE SYSTEMS
Equipotential surfaces of various charge systems. For the various charge systems, we represent equipotential surfaces by dashed curves and lines of force by full line curves. Between any two adjacent equipotential surfaces, we assume a constant potential difference.
Importance of equipotential surfaces. Like the lines of force, the equipotential surfaces give a visual picture of both the direction and the magnitude of field in a region of space. If we draw equipotential surfaces at regular intervals of V, we find that equipotential surfaces are closer together in the regions of strong field and farther apart in the regions of weak field. Moreover, is normal to the equipotential surface at every point.
ELECTRIC POTENTIAL ENERGY
Electric potential energy. It is the energy possessed by a system of charges by virtue of their positions. When two like charges lie infinite distance apart, their potential energy is zero because no work has to be done in moving one charge at infinite distance from the other. But when they are brought closer to one another, work has to be done against the force of repulsion. As electrostatic force is a conservative force, this work gets stored as the potential energy of the two charges.
The electric potential energy of a system of point charges may be defined as the amount of work done in assembling the charges at their locations by bringing them in, from infinity.