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Potential due to an Electric Dipole at a certain distance is the sum of the potentials due to both the charges of the dipole at that point. The electric potential is explained by the work required to move a unit of positive charge from a reference point to a particular point within an electric field having no acceleration.
- A dipole is referred to as a pair of opposite charges having equal magnitudes that are separated by a distance, d.
- The electric potential is explained by a scalar field where the gradient becomes the electrostatic vector field.
- Since it is a scalar field, it becomes quite easy to calculate the potential due to a system of charges.
- It is the summation of the electric potentials at a particular point in time mainly due to individual charges.
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Key Terms: Electric Dipole, electric field, acceleration, magnitudes, electric potential, scalar field, electrostatic vector field, dipole
Potential Due to an Electric Dipole
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The electric potential is explained by the work required to move a unit of positive charge from a reference point to a particular point within an electric field having no acceleration. A dipole is referred to as a pair of opposite charges having equal magnitudes that are separated by a distance, d.
The electric potential due to a point charge q at a distance of r from that charge is mentioned by:
V = q/(4πε0 r)
Where ε0 is the permittivity of free space.
The electric potential is explained by a scalar field where the gradient becomes the electrostatic vector field. Since it is a scalar field, it becomes quite easy to calculate the potential due to a system of charges. It is the summation of the electric potentials at a particular point in time mainly due to individual charges. The formula for electric potential due to an electric dipole is given by
\(V = \frac {pcos\theta}{4\pi \epsilon_o (r^2-l^2cos^2\theta)}=\frac {pcos\theta}{4\pi \epsilon_or^2}\)
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Derivation of Electric Potential due to an Electric Dipole
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Consider any point P at a distance r from the center (O) of the electric dipole AB. Let OP makes an angle θ with the dipole moment p. Let r1 and r2 be the distances of point P from -q charge and +q charge of the dipole respectively.
Potential at P due to -q charge
V1 = 1/4πε0 (-q/r1)
Potential at P due to +q charge
V2 = 1/4πε0 (+q/r2)
The net electric potential at point P due to the electric dipole is
V = V1 + V2
⇒ V = 1/4πε0 (-q/r1) + 1/4πε0 (+q/r2)
⇒ V = q/4πε0 (1/r2 – 1/r1)
Now r1 = Ap ≅ CP = OP + OC = r + l cosθ
Also r2 = BP = DP = OP – OD = r – l cosθ
Hence
V = q/4πε0 [1/(r – l cosθ) – 1/r + l cosθ)]
⇒ V = q/4πε0 [( r + l cosθ – r + l cosθ)/ (r2 – l2 cos2θ)]
⇒ V = q/4πε0 [2l cosθ / (r2 – l2 cos2θ)]
⇒ V = 2ql cosθ / 4πε0 (r2 – l2 cos2θ)
But 2ql = p (dipole moment)
⇒ V = p cosθ / 4πε0 (r2 – l2 cos2θ)
If point P is far away from the center of the dipole i.e r >> l, then
\(V = \frac {pcos\theta}{4\pi \epsilon_o (r^2-l^2cos^2\theta)}=\frac {pcos\theta}{4\pi \epsilon_or^2}\)
From the above equation, it is quite clear that potential due to the electric dipole is inversely proportional to r2 not 1/r which is the case for potential due to a single charge.
The potential due to electric dipole is not only dependable on R rather it is also dependable on the angle between position vector R and dipole moment p.
Electric Field: Electric Field means the force experienced per unit positive test charge at a point when another charge is kept in the vicinity.
Electric Dipole and Electric Dipole Moment: Electric Dipole means a system of two equal in magnitude opposite changes which are segmented by very less distance. Electric Dipole Moment is the product of charge and separation; it is considered as a vector quantity calculated by:
Electric Dipole and Electric Dipole Moment
On Axial Point:Electric field due to dipole on the axial point P is mentioned by
On Equatorial Plane: Electric field mainly because of the dipole on the equatorial plane at a point P will be mentioned as
Torque experienced by a dipole in the same electric field will be marked as
We all know that electric dipole means the +q and -q that are separated by a small distance 2a. The Electric dipole moments are mentioned by a vector P of magnitude 2qa and this vector will be in the direction ranging from negative q to positive sq.
To get an electric potential due to a dipole you can consider charge -q is placed at point P and charge +q is placed at point Q as shown below in the figure.
Since electric potential works on the superposition principle, the potential regarding the electric dipole as a whole will be the total potential due to both the charges +q and -q. The numerical term which will represent the same mentioned is:
In this equation, r1 and r2 respectively are considered the distance of charge +q and -q from point R.
You are supposed to draw line PC perpendicular to RO and line QD perpendicular to RO.
since the magnitude of the dipole is
|p| = 2qa
- If we consider the case where r>>a then again since pcosθ will be equal to p·rˆ where rˆ will be considered as the unit vector along the vector OR then the electric potential of the dipole will be defined for r>>a
Some important points need to be remembered
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- The potential due to an electric dipole of important points falls as 1/r2 and the potential due to a single point charge falls as 1/r. The potential due to the dipole falls is much more than a monopole (point charge). With a surge in distance from the electric dipole, the effects of positive and negative charges will nullify each other.
- The potential due to a point charge is spherically symmetric since it depends only on the distance r. But the potential due to a dipole will not be spherically symmetric as the potential is dependent on the angle between the position vector of the point.
- However, the dipole potential is axially symmetric. In case the position vector gets rotated about by making θ at the fixed position, then all points on the cone at the same distance r will have the same potential.
Electrostatic Potential and Capacitance is an important section including Potential due to electric dipole and system of charges.
Things to Remember
- The Physic chapter holds significant weightage to the numerical and direct derivations, so it is advised that students must not miss any important topic.
- Derivations or topics such as the energy stored in the capacitors, potential due to electric dipole, and potential energy of the system of charges are asked more in the exam.
- Both Short Answers and Long Answer questions related to numbers are asked in the exam.
- Since complex combinations of capacitors will be mentioned in the exam to evaluate the effective capacitance or charge on plates, all the students are advised to do intensive study of the topic.
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Sample Questions
Ques. What is the electric potential due to an electric dipole at an equatorial point? (1 mark)
Ans. Zero, because the potential on equatorial points, mainly due to charges of electric dipole, will be equal in magnitude but opposite in nature. Due to this the result would be zero.
Ques. A short electric dipole has a dipole moment of 4×10−94×10−9 Cm. You need to find the following? (3 marks)
(a) Electric Potential at a point distant, 3m from center of the dipole on the axial line
(b) Electric Potential at a point distant 1 m from center of the dipole on the equatorial line
(c) Electric Potential at a point distant it will be 3m from center of the dipole on an line making a 30 degree angle on the dipole axis
Ans. p=4×10−94×10−9 Cm
Potential of dipole is given as
V=14πε0pcosθr2V = 14πε0pcosθr2
(a) θ=00θ=00, r=.3m
V=14πε0pr2 = 400VV = 14πε0pr2=400V
(b) θ=900θ=900, r = 1m
V = 0V = 0
(c) θ=300θ=300, r=.3m
V=14πε0pcosθr2V=14πε0pcosθr2
Substituting the values
V=200√3V=2003 V
Ques. Is there any relationship between electric potential and electric potential energy? (2 marks)
Ans. Electric potential energy means the potential energy stored when charges are out of equilibrium (like gravitational potential energy). The electric potential will remain the same, but per charge, Ueq. An electric potential difference between two points is referred to as voltage, V=Ue2q−Ue1q.
Ques. What is the potential due to an electric dipole? (2 marks)
Ans. To get the electric potential due to a dipole consider charge -q is placed at point P and charge +q is placed at point Q as shown below in the figure. Potential due to electric dipole will not only be dependable on r but will also depend on the angle between position vector r and dipole moment p.
Ques. What is the maximum electric potential due to an electric dipole? (2 marks)
Ans. Explanation: The potential energy of an electric dipole is the product of the dipole moment and electric field. When the angle between the dipole moment and electric field is 180° then the potential energy of the electric dipole is maximum
Ques. An electric field of 1000 V/m will be applied at an electric dipole at 45° angle; its value is 10–29 Cm. You need to calculate the potential energy of the electric dipole? (3 marks)
(a) –10 × 10–29 J
(b) –7 × 10–27 J
(c) –20 × 10–18 J
(d) –9 × 10–20 J
Ans. E will be equal to 1000 V/m, p would be equal to 10-29 cm, θ will be a 450 angle
The potential energy stored in the dipole,
U will be equal to p.Ecos θ = 10-29 x 1000 x cos450
U will be equal to \frac{-1}{\sqrt{2}}\times 10^{-26}2−1×10−26
U will be equal to– 0.707 x 10-26 J= -7 x 10 -27 J
Ques. An electric dipole will be kept at an angle of 30 degrees. What will be the dipole experience? (2 marks)
(a) it will be only torque
(b) it will be a translational force in the direction of the field
(c) it will be a translational force in the normal direction to the direction of the field
(d) both torque and translational force
Ans. In a non-uniform electric field, the dipole will experience a torque and a translational force.
Ques. A charge q is moved from point A above a dipole of dipole moment p to point B below the dipole in an equatorial plane without acceleration. Find out the work done in the process. (All India 2016) (2 marks)
Ans. No work is done when a charge q is moved from a point A above a dipole of dipole movement p to a point B below the dipole in an equatorial plane without acceleration.
[W = q VAB = q x 0 = 0, as the potential remains constant].
Ques. Obtain the expression for the electric potential at any point along the axial line of an electric dipole. (5 marks)
Ans. Let us consider an electric dipole consisting of two points charged -q and +q and separated by a distance 2a. Let P be a point on the axis of the dipole at a distance r from its O.
Due to dipole, electric potential at point P is,
V = V1 + V2
Ques. AB and CD are two small identical electrical dipoles where each dipole moment p is kept at an angle of 120 degree as shown in the figure. What X’ is the resultant dipole moment of this combination? If this system is subjected to an electric field E directed along +X direction, what will be the magnitude and direction of the torque acting on this? (Delhi 2011) (5 marks)
Ans. The resultant dipole moment of both dipoles is,
The resultant dipole moment p makes an angle of 60 degree with each dipole and 30 degree with x-axis as shown in the figure above.
Ques. An electric dipole is held in a uniform electric field.
(i) Show that the net force acts on it is zero.
(ii) The dipole is aligned parallel to the field.
Obtain the work done in rotating it through the angle of 180 degree. (All India 2012) (5 marks)
Ans. (i) Due to charge -q, force acting on point A is -qE
Due to charge +q, force acting on B is +qE
Thus, net force acting on,
-qE +qE = (zero)
Therefore, the net force acting on an electric dipole held in a uniform electric field is zero.
(ii) W = -pE(cos 02 - cos θ2)
W = -pE(cos 180 degree - cos 0°)
W = -pE(-1 -(1)) = +2pE.
Ques. (i) Obtain the expression for the electric potential due to an electric dipole at a point on its axial line.
(ii) Depict the equipotential surfaces due to an electric dipole. (Delhi 2015) (5 marks)
Ans. (i) The electric potential due to an electric dipole is the sum of potentials due to the charges q and -q
…. (i)
Here r1 and r2 are the distances of the point P from q and -q respectively.
Now, by geometric method,
By using the binomial theorem and retaining terms upto the first order in a/r, we get
Where 
is the unit vector along the position vector OP.
Then the electric potential of dipole is given by
… (iv)
From the above equation (iv), potential on the dipole axis (θ = 0, n) is given by,
(ii) Equipotential surface for an electric dipole
Ques. (a) Find the expression for the potential due to an electric dipole of dipole moment p at a point V on the axial line.
(b) Two identical capacitors of plate dimensions l x b and plate separation d have dielectric slabs filled in between the space of the plates as shown in the figure below.
Derive the relation between the dielectric constants K, K1 and K2. (Comptt. All India 2013) (10 marks)
Ans. (a) Consider an electric dipole consisting of two equal and opposite charges -q at A and +q at B which is separated by a distance 21 with centre at O. we have to calculate potential at a point P whose polar coordinates are (r, 0) that is OP = r and ∠BOP = θ.
Here AP = r1 and BP = r2, thus we can calculate potential as P due to point charges at A and B using V is
The potential at P due to both the charges of the dipole is V = V1 + V2
.... (i) In order to put this result in a more convenient way, we can draw normals form A and B on the line joining O and P.
From ΔBOD, OD = I cos θ and from ΔOAC, OC = I cos θ
From the figure we get, PB = PD and PA = PC
Thus, r1 = r + 1 cos θ,
Using these outcomes in equation 1 we get,
In terms of dipole (p = q x 21), we can express this result as
... (ii)
The outcome shows that unlike potential due to a dipole is inversely proportional to the square r of the distance.
Now consider its special cases,
Case I- when the point P lies on the axial line of the dipole on the side of positive charge, θ = 0 and cos θ = 1
Then equation 2 reduces to
... (iii)
Case II- when the point P lies on the axial line of the dipole but on the side of negative charge, θ = 180 degree and cos θ = 1
... (iv)
Case III- when point P lies on the equatorial line of the dipole, θ = 90 degree and cos θ = 0
Then V equatorial = 0 … (i)
Hence, electric potential due to a dipole is zero at all the points on the equatorial line of the dipole.
(b) In the first case,
... (i)
In the second case, these two apartments are in parallel and their net capacity would be the sum of two individual capacitances.
As these are identical capacitors, comparing (i) and (ii)
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