Electric Dipole
If two charges of equal magnitude and opposite nature are present at a very small distance from each other, they form an electric dipole.
If two charges +q and - q are situated at a distance `2 l` (may be 2r or 2a).
`star` The midpoint between the two charges is called the center of electric dipole.
`star` The line joining these charges is called axial line.
`star` The line perpendicular to axial line and passing through the center is called the equatorial line.
Potential Energy of a Dipole in an External Field
The potential energy of an electric dipole in an electric field is equal to the work that is done to bring an electric dipole from infinity to that field.
Consider AB be an electric dipole is brought from infinity to in a uniform electric field `\vecE` so that the dipole moment `\vec P` is always along electric field `\vec E`.
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| Electric Dipole |
The force on charge +q due to electric field `\vec E` along the field and force on charge - q is opposite to the field. Thus external work is done for bringing charge q of dipole in electric field where as the electric field itself work on charge -q. On coming from infinifity to electric field, -q charge has to move 2a more distance that by q. Thus, the work done by the charge -q is more negative. Thus, work done by the electric field.
`W = \text{force on charge (-q) \times \text{distance covered}}`
`W = - qE \times 2a `
`W = - q (2a) \times E`
`W = - p E` `(\because q (2a) = p)`
Thus, the potential energy of electric dipole placed paralleled to the electric field `\vec E`
`U_1 = - p E`
Now, work done in rotating by angle `\theta` from the parallel position of the electric field `\vec E`
`U_2 = p E ( 1 - cos\theta)`
Thus, potential energy of electric dipole at angle `\theta` in electric field,
` U = U_1 + U_2`
` U = - p E + p E (1 - cos\theta) `
` U = - p E + p E - p E cos\theta) `
` U = - p E cos\theta `
In vector form,
`U = - \vec p. \vec E`
Special Cases
Case I
If the electric dipole moment is at `0^\circ` angle from the electric field, then the potential energy
`U = - p E cos\theta`
`U = - p E cos0^\circ`
`U = - p E (1)`
`U = - p E` (Stable equilibrium)
It is a stable equilibrium because in this case, the energy is minimum.
Case II
If the electric dipole moment is perpendicular to the electric field, then the potential energy
`U = - p E cos\theta`
`U = - p E cos90^\circ`
`U = - p E (0)`
`U = 0`
Case III
If the electric dipole moment is at `180^\circ` angle from the electric field, then the potential energy
`U = - p E cos\theta`
`U = - p E cos180^\circ`
`U = - p E ( - 1)`
`U = + p E` (Unstable equilibrium)
It is an unstable equilibrium because in this case, the energy is maximum.
Solved Example
An electric dipole of charges` +1.0 \times 10^{-6} C` and
` -1.0 \times 10^{-6} C` are at a distance of 2 cm. It is placed in a uniform electric field of`1.0 \times 10^5 V/m`. Determine :
(a) Maximum torque acting on the dipole by the electric field.
(b) Potential energy in the stable condition of equilibrium.
(c) Potential energy in rotating the dipole by `180^\circ` from the stable equilibrium condition.
(d) Essential energy to rotate the dipole by `90^\circ` along the electric field.
Given
` q =1.0 \times 10^{-6} C`
`2a = 2 cm = 2 \times 10^{-2}`
`E = 1.0 \times 10^5 V/m`
Dipole, moment,
`p = q (2a)`
`p = 1 \times 10^{-6} (2 \times 10^{-2})`
`p = 2 \times 10^{-8} N m`
(a) Maximum torque,
`\tau = p E`
`\tau = 2 \times 10^{- 8} \times 1 \times 10^5`
`\tau = 2 \times 10^{- 3} N m`
(b) Potential energy in stable equilibrium
`U = - p E`
`U = - 2 \times 10^{- 8} \times 1 \times 10^5`
`U = - 2 \times 10^{- 3} J`
(c) Potential energy in rotating by `180^\circ` from stable equilibrium
`U = p E`
`U = 2 \times 10^{- 8} \times 1 \times 10^5`
`U = 2 \times 10^{- 3} J`
(d) Work done in rotating by `90^\circ`
`W = p E (1 - cos\theta)`
`W = 2 \times 10^{- 8} \times 1 \times 10^5 (1 - cos90^\circ)`
`W = 2 \times 10^{- 8} \times 10^5 (1 - 0)`
`W = 2 \times 10^{- 3} (1)`
MCQs with answer
1. What is an electric dipole?
(a) A single charge
(b) A pair of opposite charges
(c) A charged sphere
(d) A charged conductor
2. Electric dipole moment of an electric dipole depends on:
(a) The magnitude of the charges
(b) The distance between the charges
(c) The orientation of the charges
(d) All of the above
3. The electric dipole moment is a:
(a) Vector quantity
(b) Scalar quantity
(c) Both vector and scalar quantity
(d) None of the above
4. The potential due to an electric dipole at a point on its equatorial line is:
(a) Proportional to the inverse of the distance from the dipole
(b) Proportional to the square of the distance from the dipole
(c) Zero
(d) Independent of the distance from the dipole
5. The electric field due to an electric dipole is zero at:
(a) Points on the axial line
(b) Points on the equatorial line
(c) Points on the perpendicular bisector of the dipole
(d) All points in space
6. If an electric dipole is placed in uniform electric field, it experiences :
(a) A torque
(b) A force
(c) Both torque and force
(d) No interaction
7. The torque experienced by an electric dipole in a uniform electric field is maximum when:
(a) Dipole is parallel to the electric field
(b) Dipole is perpendicular to the electric field
(c) Dpole is at an angle of 45 degrees with the electric field
(d) The torque does not depend on the angle between the dipole and the electric field
8. Potential energy of an electric dipole in a uniform electric field is minimum when:
(a) Dipole is parallel to the electric field
(b) Dipole is perpendicular to the electric field
(c) Dipole is at an angle of 45 degrees with the electric field
(d) Potential energy does not depend on the angle between the dipole and the electric field
Answer
1. What is an electric dipole?
Answer: (b) A pair of opposite charges
2. Electric dipole moment of an electric dipole depends on:
Answer: (d) All of the above
3. The electric dipole moment is a:
Answer: (a) Vector quantity
4. The potential due to an electric dipole at a point on its equatorial line is:
Answer: (c) Zero
5. The electric field due to an electric dipole is zero at:
Answer: (b) Points on the equatorial line
6. (c) Both torque and force
The electric field exerts a force on the positive and negative charges of the dipole in opposite directions, resulting in a torque that tends to align the dipole with the electric field. At the same time, the charges experience equal and opposite forces, leading to a net zero force on the dipole as a whole. Therefore, the dipole experiences both torque and force in this situation.
7. The torque experienced by an electric dipole in a uniform electric field is maximum when:
The correct answer is (b) Dipole is perpendicular to the electric field.
Explanation:
The torque is a measure of the rotational force acting on the dipole. The magnitude of the torque experienced by an electric dipole can be calculated using the formula:
τ = pEsinθ
Where:
`\tau` is the torque experienced by the dipole,
p is the magnitude of the dipole moment,
E is the electric field,
θ is the angle between the dipole moment vector and the electric field vector.
From the formula, we can observe that the torque is directly proportional to the sine of the angle between the dipole moment and the electric field. The maximum value of the sine function is 1, which occurs when the angle is 90 degrees (perpendicular). This means that the torque experienced by the dipole is maximum when the dipole is perpendicular to the electric field.
Therefore, option (b) Dipole is perpendicular to the electric field is the correct answer.
8. Potential energy of an electric dipole in a uniform electric field is minimum when:
Answer: (a) Dipole is parallel to the electric field
Other Useful Digram
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| Electric Dipole |
Q. What is a Dipole?
A dipole is a pair of equal and opposite charges separated by a very small distance. The positive and negative charges within the dipole create an electric field between them.
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