**Electrostatic Potential and Capacitance Important Questions for CBSE Class 12 Physics Electrostatic Potential**

**1.Electrostatic Potential** The electrostatic potential at any point in an electric field is equal to the amount of work done per unit positive test charge or in bringing the unit positive test charge from infinite to that point, against the electrostatic force without acceleration.

**NOTE** Electrostatic potential is a state dependent function as electrostatic forces are conservative forces

**2.Electrostatic Potential Difference** The electrostatic potential difference between two points in an electric field is defined as the amount of work done in moving a unit positive test charge from one point to the other point against of electrostatic force without any acceleration (i.e. the difference of electrostatic potentials of the two points in the electric field).

where, is work done in taking charge q_{0} from A to B against of electrostatic force.

Also, the line integral of electric field from initial position A to final position B along any path is termed as potential difference between two points in an electric field, i.e.

**NOTE** As, work done on a test charge by the electrostatic field due to any given charge configuration is independent of the path, hence potential difference is also same for any path.

For the diagram given as below, potential difference between points A and B will be same for any path.

**3**.Electrostatic potential due to a point charge q at any point P lying at a distance r from it is given by

**4.**The potential at a point due to a positive charge is positive while due to negative charge, it is negative.

**5.**When a positive charge is placed in an electric field, it experiences a force which drives it from points of higher potential to the points of lower potential. On the other hand, a negative charge experiences a force driving it from lower potential to higher.

**10.**Graphical representation of variation of electric potential due to a charged shell at a distance r from centre of shell is given as below

**11.Equipotential Surface** A surface which have same electrostatic potential at every point on it, is known as equipotential surface.

The shape of equipotential surface due to

(i) line charge is cylindrical.

(ii) point charge is spherical as shown along side:

- Equipotential surfaces do not intersect each other as it gives two directions of electric field E at intersecting point which is not possible.
- Equipotential surfaces are closely spaced in the region of strong electric field and vice-versa.
- Electric field is always normal to equipotential surface at every point of it and directed from one equipotential surface at higher potential to the equipotential surface at lower potential.
- Work done in moving a test charge from one point of equipotential surface to other is zero.

where, negative sign indicates that the direction of electric field is from higher potential to lower potential, i.e. in the direction of decreasing potential.

**NOTE** (i) Electric field is in the direction of which the potential decreases steepest.

(ii) Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface at the point.

**13.** **Electrostatic Potential Energy** The work done against electrostatic force gets stored as potential energy. This is called electrostatic potential energy

**14.**The work done in moving a unit positive test charge over a closed path in an electric field is zero. Thus, electrostatic forces are conservative in nature.

**15.**Electrostatic potential energy of a system of two point charges is given by

Putting the values of charge with their signs.

**16**.Electrostatic potential of a system of n point charges is given by

**17.Potential Energy in an External Field**

(i)**Potential Energy of a single charge in external field** Potential energy of a single charge q at a point with position vector r, in an external field is qV(r),where V(r) is the potential at the point due to external electric field E.

(ii)**Potential Energy of a system of two charges in an external field**

where, q_{x} and q_{2} – two point charges at position vectors r_{:} and r_{2}, respectively

V (r_{x}) = potential at r_{x} due to the external field

V(r_{2}) = potential at r_{2} due to the external field

**18**.Potential energy of a dipole in a uniform electric field E is given by Potential energy = -p .E

**19.Electrostatic Shielding** The process which involves the making of a region free from any electric field is known as electrostatic shielding

It happens due to the fact that no electric field exist inside a charged hollow conductor. Potential inside a shell is constant. In this way we can also conclude that the field inside the shell (hollow conductor) will be zero

**Previous Year Examination Questions**

**1 Mark Questions**

**1.The figure shows the field lines of a positive charge. Is the work done by the field is moving a small positive charge from Q to P positive or negative? [Foreign 2014]**

** **

**Ans.**

**2.For any charge configuration,equipotential surface through a point is a normal to the electric field. Justify. [Delhi 2014]**

**Ans.**

**Ans.**

**4.Why electrostatic potential is constant throughout the volume of the conductor and has the same value as on its surface? [Delhi 2012]**

**Ans**. Since, electric field intensity inside the conductor is zero. So, electrostatic potential is a constant.

**5.Why is the potential inside a hollow spherical charged conductor is constant and has the same value as on its surface?[Foreign 2012]**

**Ans**.Electric field inside the hollow spherical charged conductor is zero. So, no work is done in moving a charge inside the shell.

This implies that potential is a constant and therefore, equal to its value at the surface, i.e.

**6.Why there is no work done in moving a charge from one point to another on an equipotential surface? [Foreign 2012]**

**Ans.**

**7.A hollow metal sphere of radius 5 cm is charged such that potential on its surface is 10 V. What is the potential at the centre of the sphere? [All India 2011]**

**Ans.**

**8.Can two equipotential surface intersect each other? Justify your answer. [Delhi 2011 c]**

**Ans**.(i) No, two equipotential surfaces cannot intersect each other because two normals can be drawn at intersecting point on two surfaces which give two directions of E at the same point which is impossible.

(ii) Also, two values ot potential at the same point is not possible.

**9.A point charge Q is placed at point O as shown in the figure. Is the potential difference (V _{A} – V_{B})**

**positive, negative or zero if Q is (i)positive (ii) negative**

**Ans.**

**10.Draw equipotential surfaces due to a single point charge. [All India 2011 c]**

**Ans.**Equipotential surfaces due to a single point charge are concentric sphere having charge at the centre.

**11.Name the physical quantity whose SI unit is J/C. Is it a scalar or a vector quantity? [All India 2010]**

**Ans.** J/C is the SI unit of **electric **potential. It is a scalar quantity.

**12.What is the work done in moving a test charge q through a distance of 1 cm along the equatorial axis of an electric dipole? [All India 2009]**

**Ans.**

**13.Define the term potential energy for charge q at a distance r in an external field. [All India 2009]**

**Ans**.The electric potential energy at any point lying at a distance **r** from the source charge **q** is equal to the amount of work done in moving unit positive test charge from infinity to that point without any acceleration against of electrostatic force.

**14.The potential due to a dipole at any point on its axial line is zero. [All India 2009 C]**

**Ans.**Wrong, as potential due to an electric dipole is zero on equatorial line in spite of axial line.

The potential due to a dipole at any point on equatorial line is zero, not an axial line.

**15.What is the electric potential due to an electric dipole at an equatorial point?[All India 2009]**

**Ans.**Zero, as potential on equatorial point due to charges of electric dipole, is equal to magnitude but opposite in nature and hence, their resultant is zero

**Ans.**

**2 Marks Questions**

**17.Two point charges q _{1} and q_{2} are located at q and r_{2}, respectively in an external electric field E. Obtain the expression for the total work done in assembling this configuration. [Delhi 2014 C]**

**Ans.**

**Ans.**

**19.A test charge q is moved without acceleration from A to C along the path from A to B and then from B to C in electric field E as shown in the figure.**

** (i)Calculate the potential difference between A and C.**

** (ii)At which point (of the two) is the electric potential more and why?**

** **

**Ans.**

**20.Graph the electric potential (V) with distance r due to a point charge Q.[Delhi 2012]**

**Ans.**

**Ans.**

**22.A dipole with its charge -q and +q located at the points (0, – b,0) and (0,+ 5,0) is present in a uniform electric field E. The equipotential surfaces of this field are planes parallel to the Y Z-plane.**

** (i)What is the direction of the electric field E?**

** (ii)How much torque would the dipole experience in this field?[Delhi 2010 C]**

**Ans.**

**Ans.**

**24.Find out the expression for the potential energy of a system of three charges q _{1}, q_{2} and g_{3} located at q , r_{2} and r_{3} with respect to the common origin O. [Delhi 2010 c]**

**Ans.**

**25.Draw three equipotential surfaces corresponding to a field that uniformly increases in magnitude but remains constant along Z-direction. How are these surfaces different from that of a constant electric field along Z-direction? [Foreign 2008; All imiia 2009]**

**Ans.**

In case of constant electric field along Z-direction,the perpendicular distance between equipotential surfaces remains same whereas for field of increasing magnitude, equipotential surfaces get closer as we go away from the origin.In both cases, surfaces be planes parallel to **XY-** plane.

**26.(i)Can two equipotential surfaces intersect each other? Give reasons,**

** (ii)Two charges – qand + qare located at points A (0, 0, – a) and B{0, 0, +a) respectively. How much work is done in moving a test charge from point P{7, 0, 0) to Q(- 3, 0, 0)? [Delhi 2009]**

**Ans**.(i) No, two equipotential surfaces cannot intersect each other because two normals can be drawn at intersecting point on two surfaces which give two directions of E at the same point which is impossible.

**27.(i) Can two equipotential surfaces intersect each other? Give reasons, (ii) Two charges + q and – q are located at points A(0, 0, – 2) and B (0, 0, 2) respectively. How much work will be done in moving a test charge from point P{7,0,0) to Q (- 3, 0, 0)? [Delhi 2009]**

**Ans**.(i) No, two equipotential surfaces cannot intersect each other because two normals can be drawn at intersecting point on two surfaces which give two directions of E at the same point which is impossible.

**28.(i)Write two characteristics of equipotential surfaces.**

** (ii) Draw the equipotential surfaces due to an electric dipole. [All India 2009 C]**

**Ans.**(i) Two equipotential surfaces do not intersect each other as normals at intersecting points on two surfaces will give two directions of electric field which is impossible.

(ii) Closely spaced equipotential surfaces represent strong electric field and vice-versa.

**29.Two point charges, q _{t} =10xl0^{-8} C and q_{2} =-2×10 ~^{8} C are separated by a distance of 60 cm in air.**

**(i)What a distance from the 1st charge q**

_{x}would the electric potential be zero?**(ii)Also, calculate the electrostatic potential energy of the system.[All India 2007,2008]**

**Ans.**

**30.Two point charges 40, O are separated by lm in air. At what point on the line joining the charges, is the electric field intensity zero? Also calculate the electrostatic potential energy of the system of charges taking the value of charge, O = 2 X 10~ ^{7} C. [All India 2008]**

**Ans.**

**31.Define the dipole moment of an electric dipole. How does the electric potential due to a dipole vary on the dipole axis as a function of r distance of the point from the mid-point of the dipole at large distances? [All India 2008 C]**

**Ans**.Electric field inside the hollow spherical charged conductor is zero. So, no work is done in moving a charge inside the shell.

This implies that potential is a constant and therefore, equal to its value at the surface, i.e.

variation of electric potential due to dipole on axial lines when point is at large distance from mid-point of the dipole, occurs as per relation below.

The variation of electric potential with distance r is shown as below:

**32.Derive an expression for the electric potential at any point along the axial line of an electric dipole. [Delhi 2008]**

**Ans.**

**33.Two charges of 5 nC and – 2 nC are placed at points (5 cm, 0, 0) and (23 cm, 0, 0) in the region of space, where there is no other external field. Calculate the electrostatic potential energy of this charge system.[Delhi 2008 C]**

**Ans.**

**3 Marks Questions**

**34.A wire AB is carrying a steady current of 12 A and is lying on the table. Another wire CD carrying 5 A is held directly above AB at a height of 1mm. Find the mass per unit length of the wire CD, so that it remains suspended at its position when left free. Give the direction of the current flowing in CD with respect to that in [Take the value of g =10 ms ^{-2}][All India 2013]**

**Ans.**

**35.(i) Depict the equipotential surfaces for a system of two identical positive point charges placed a distance d apart.**

** (ii) Deduce the expression for the potential energy of a system of two point charges q _{1} and q_{2} brought from infinity to the points with positions and r_{2} respectively, in presence of external electric field E. [Delhi 2010]**

**Ans.**

By definition, electric potential energy of any charge q placed in the region of electric field is equal to the work done in bringing charge q from infinity to that point and given by U =qV

where, V is the electric potential (as potential at infinity is assumed to be zero) where, the charge q is placed.Now, considering the electric potentials at positions r1 and r

_{2}as V1, and V

_{2}, respectively. Therefore, total potential energy of the system of two charges q•, and q

_{2}placed at points with position vectors r, and r

_{2}in the region of E is given by

**U=**work done in bringing charge

**q**from infinite to that position in E is equal to work done for charge

**q**from infinite to that position in E +work done to that of charge

_{2}**q**at these positions in presence of

_{2}**q1**

**Ans.**

**37.(i) Two points charges q _{x} and q_{2} initially at infinity, are brought one by one to points P_{x }and P_{2} specified by position vectors r_{t} and r_{2} relative to same origin. What is the potential energy of this charge configuration? Define an equipotential surface.**

**(ii) Draw schematically the equipotential surface corresponding to a field that uniformly increases in magnitude but remains constant in direction. [Delhi 2008]**

**Ans.**

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