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Electric potential is the amount of energy needed to move a unit charge from one point to another. It is the energy that is needed to move the charge against the electric field. Electric potential due to a point charge is the amount of work done to move a unit charge from infinity to that point against the electric field. Thus, electric potential due to a point charge is,
\(v = \frac{kQ}{r}\)
Where,
V = Electric potential,
Q = Source charge and
r = Position vector of the charge
k = Constant (\(\frac{1}{4\pi\epsilon_0}\))
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Key Terms: Electric Potential, Electric Charge, Work done, Energy, Electric Potential Energy, Electrostatic Potential, Point Charge
What is Electric Potential?
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- Electric potential is defined as the amount of energy to move a unit electric charge from a reference point to a specific point.
- Electric potential is determined by dividing the electric potential energy by the quantity of charge.
- It is also called electrostatic potential. It is a scalar quantity.
Electric potential depends on two factors,
- The electric charge the object carries
- Relative position of the object with other electrically charged objects
In turn, the strength of the electric field depends on the electric potential. It does not depend on whether a charge should be placed in the electric field or not.
Electric Potential Due to a Point Charge
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Electric potential at a point in an electric field is defined as the amount of work done in moving a unit positive charge from infinity to that point along any path when the electrostatic forces are applied. Here we will assume that the charge is positive (q>0).
\(v = \frac{q}{4\pi\epsilon_0r}\)
Where,
V is the electric potential,
q is the source charge (positive), and
r is the position vector of the positive charge.
The electric potential due to a point charge is given by,
\(v = \frac{kQ}{r}\)
Here,
k is a constant and is equal to \(\frac{1}{4\pi\epsilon_0}\).
The SI unit of electric potential is Volt (V), i.e. 1 V = 1 Joule Coulomb-1(JC-1)
Electric Potential Due to a Point Charge
Also Read:
| Laws & Concepts Related to Electric Potential | ||
|---|---|---|
| Gauss Law | Coulomb’s law | Dipole in a Uniform External Field |
| Electric Dipole | Electric charges and field | Electric Flux |
| Electric Field Formula | Unit of Electric charge | Charge density |
Electric Potential due to a System of Charges Derivation
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When we are dealing with a system of multiple charges, we can find the electric potential at an individual point by algebraically adding all the potentials due to each individual charge. Let us consider a group of charges as q1, q2, q3 with position vectors r1, r2, r3 respectively.
The electric potential V1 with charge q1 and position vector r1 will be denoted as:
\(V_1 = \frac{q_1}{4\pi\epsilon_0r_1}\)
Electric potentials V2, V3,...and so on will be denoted in a similar manner. The potential V at one point due to the system of charges will be derived by adding them, i.e,
Things to Remember
- Electric Potential is the energy required to move a unit charge from one point to another against the electric field.
- Electric potential due to a point charge is given by, \(v = \frac{q}{4\pi\epsilon_0r}\) or \(v = \frac{kQ}{r}\)
- By definition, the amount of work energy to move a unit electric charge from a reference point to a specific point is called electric potential or electrostatic potential.
- Electric potential is a scalar quantity.
-
Electric potential depends on the charge and relative position.
Previous Year Questions:
- Two long current carrying thin wires, both with current I, are held by insulating threads of length…? [JEE Mains 2015]
- Let a total charge 2Q be distributed in a sphere of radius R, with the charge density …? [JEE Mains 2019]
- A is the area of Gaussian surface and qenc is charge enclosed by the Gaussian surface. This equation can…? [JEE Mains 2020]
- For a uniformly charged ring of radius R, the electric field on its axis has the largest…? [JEE Mains 2019]
- Charge is distributed within a sphere of radius R with a volume charge density…? [JEE Mains 2019]
- An electric dipole has a fixed dipole moment p, which makes angles θ with respect to x-axis…? [JEE Mains 2017]
- A thin disc of radius b=2a has a concentric hole of radius 'a' in it…? [JEE Mains 2015]
- A spherically symmetric charge distribution is characterised by a charge density having the following…? [JEE Mains 2014]
- A particle of charge q and mass m is subjected to an electric field…? [JEE Mains 2020]
- A long cylindrical shell carries positive surface charge σ in the upper half and negative…? [JEE Mains 2015]
- The region between two concentric spheres of radii ′a′ and ‘b′, respectively…? [JEE Mains 2016]
- A body of mass M and charge q is connected to a spring of spring constant k. It is oscillating along…? [JEE Mains 2018]
- Consider a sphere of radius R which carries a uniform charge density ρ. If a sphere of…? [JEE Mains 2020]
- An electric dipole is formed by two equal and opposite charges q with separation d. The charges…? [JEE Mains 2019]
- Two charges, each equal to q, are kept at x=−a and x=a on the x−axis. A particle of mass m…? [JEE Mains 2013]
- Two parallel metal plates having charges +Q and −Q face each other at a certain distance between them…?
- A charge +q is placed at origin. There are two concentric spherical surfaces; S1 of radius…?
- A charge Q is distributed uniformly in a sphere (solid). Then, the electric field at any point…?
- If a charged spherical conductor of radius 10 cm has potential V at a point distant 5 cm…?
- Two identical spheres carrying charges - 9μC and 5μC respectively are kept in contact…?
Sample Questions
Q1. Find the electric field at any point (x, y, z) in the space given by electric potential V = 4x2-3x. (3 marks)
Solution : Electric field is defined as the negative of the gradient of the potential.
Which can be written as
Where the negative sign says that electric field approaches from positive charge to negative charge, the electric potential decreases.
Expanding the gradient operator in space coordinates
Substituting the given potential, which has only the x - component,
electric field at any point (x, y, z) is
Thus, the electric field at any point (x, y, z) is .
Ques : The electric potential at a point is zero. Explain if it is possible that the electric field at that point is non-zero. (3 marks)
Solution. If there are no charges present then the electric potential will be zero and so will the electric field. However, if there are charges present then there must be at least two charges of opposite signs, or else electric potential cannot be zero.
When the charges are equal in magnitude, the electric potential will be zero anywhere on the perpendicular bisector of the line joining the two charges.
The electric field, however in such case, can be zero only when lying on the line joining the two charges and it will be non zero when lying between the two charges. Hence, if there are charges present, the electric field is non zero.
Ques: Two charges 5×10-8Cand -3×10-8C are located 16 cm apart. At what points on the line joining the two charges is the electric potential zero? Assume the potential at infinity to be zero. (3 marks)
Solution: Given, q1=5×10-8C and q2=-3×10-8C
Distance between two charges is d=16 cm=0.16 m
Case 1: A point P falls inside the system of charges
Let r be the distance between a point P and charge q1.
As per the question, V = 0
V= q1 4πε0r + q2 4πε0( d−r )
By substituting the given values, we get
0= 5×10−8 x 4π ε0 r + ( −3× 10−8 ) x 4πε0 ( 0.16−r )
5× 10−8 4π ε0r = 3× 10−8 4πε0 ( 0.16−r )
0.16 r −1= 3/5
r=0.1 m
Thus, the potential is zero at a distance of 10 cm from q1 charge.
So, the potential will be zero at a distance of 10 cm from the positive charge between the charges.
Case 2: Point P lies outside the system of charges
Let s be the distance between point P and charge q2.
V= q14πε0 r + q2 4πε0 ( r−d )
By substituting the given values, we get
0= 5×10−8 4π ε0r + (−3× 10−8 ) 4πε0 ( r−0.16 )
5× 10−8 4π ε0 r = 3× 10−8 4πε0 ( r−0.16 )
1− 0.16 r = 3/5
r=0.4 m
Thus, the potential is zero at a distance of 40 cm from q1 charge.
.So, the potential will be zero at a distance of 40 cm from the positive charge outside the multiple charges.
Ques. An infinitely long thin straight wire has a uniform linear charge density of 13cm−1. Then, the magnitude of the electric intensity at a point 18cm away is (given ε0=8.8×10−12C2Nm−2) (3 marks)
Ans. Charge density of long wire λ=½ C−m
and
r=18×10−2m
From Gauss theorem, E.dS = q/ε0
E×2πrl = q/ε0
E = q/2πε0rl
=9×109×1/3 × 2 ×1/(18×10−2)
=0.33×1011NC−1
Ques. A battery of constant voltage is available. How to adjust a system of three identical capacitors to get high electrostatic energy with the given battery? (2 marks)
Ans. Electrostatic energy of the capacitor is given by ½ CV2. Now, if three capacitors are in parallel combination, there equivalent capacitance will be maximum, So electrostatic energy of the capacitor will be maximum.
Ques. In a Millikan's oil drop experiment, the charge on an oil drop is calculated to be 6.35 x10−19C. The number of excess electrons on the drop is? (2 marks)
Ans. We know that,
Q=ne
Number of electron n= Q/e
= 6.35×10−19 /1.6×10−19
= 3.9
Ques. Two charges of equal amount +Q are placed on a line. Another charge q is placed at the mid-point of the line. The system will be in equilibrium if the value of q is? (3 marks)
Ans. For the equilibrium of the system Force on +Q charge, KQ2/ℓ2
KQq/ℓ2 x 4q= -Q/4
Ques. Calculate the surface charge density (in Cm−2) of the earth?
Ans. Charge on the earth =−0.5×106C
Charge density = charge/area
Charge density = 0.5×106/4π×(6400×103)2
=−10−9Cm−2
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