Electric Dipole MCQ Quiz - Objective Question with Answer for Electric Dipole - Download Free PDF

Explanation:

For a point outside sphere, we have,

\(E=\frac{Q}{4 \pi \varepsilon _{o} R^{2}}\)

\(1. Q \rightarrow Q, R \rightarrow R \Rightarrow E_{1}=E\)

\(2. Q \rightarrow 2Q, R \rightarrow R \Rightarrow E_{2}=2E\)

\(3. Q \rightarrow Q', R \rightarrow R \Rightarrow E_{3}=\rho R/3\epsilon_0 \\ \Rightarrow E_3=\frac{4Q}{4/3\pi (2R)^3}\frac{R}{3\epsilon_0} =E/2\)

\(\Rightarrow E_{2}>E_{1}>E_{3}\)

Explanation:

An electric dipole in a uniform electric field experiences a torque that tends to align the dipole with the electric field. The torque ( $\overrightarrow{\tau}$ ) acting on an electric dipole moment ( $\overrightarrow{P}$ ) placed in a uniform electric field ( $\overrightarrow{E}$ ) is given by the vector cross product of the dipole moment and the electric field.

The formula for the torque is:

$\overrightarrow{\tau} = \overrightarrow{P} \times \overrightarrow{E}$

The cross product of two vectors results in a vector that is perpendicular to both, and its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them.

Therefore, the correct answer is option 1: $\overrightarrow{P} \times \overrightarrow{E}$ .

∴ The torque acting on the electric dipole is $\overrightarrow{P} \times \overrightarrow{E}$ .

Concept:

Torque on an Electric Dipole in an Electric Field:

The torque (τ) acting on an electric dipole placed in a uniform electric field is given by the formula:

τ = pE sin(θ)

Where:

τ is the torque,

p is the dipole moment,

E is the magnitude of the electric field, and

θ is the angle between the dipole moment and the electric field.

Calculation:

Given:

Dipole moment, p = 4 × 10−9 C·m

Electric field, E = 5 × 104 N/C

Angle, θ = 60°

Using the formula for torque:

⇒ τ = pE sin(θ)

⇒ τ = (4 × 10−9 C·m) × (5 × 104 N/C) × sin(60°)

⇒ τ ≈ 1.73 × 10−4 N·m

∴ The magnitude of the torque acting on the dipole is 1.73 × 10−4 N·m.

Ans.(4)

Sol.

Dipole moment of each molecule = 10^-30 cm.

As 1 mole of the substance contains 6 × 10²³

molecules Total dipole moment of all the molecules

p = 6 × 10²³ × 10^-30 Cm = 6 × 10^-7 Cm

Initial potential energy U_i = pEcos θ = -6 × 10^-7 × 10⁷ cos 0° = -6]

Final potential energy U_f = -6 × 10^-6 × 10⁶ × cos 60° = -3J

Work done (Change in potential energy)

= -[(-3) - (-6)] = 3J

Concept:

For an electric dipole in a non-uniform electric field, the force and torque on the dipole depend on the alignment of the dipole moment with respect to the electric field.

Explanation: 

When the dipole moment p" id="MathJax-Element-764-Frame" role="presentation" style="position: relative;" tabindex="0">pp $\mathbf{p}$ is parallel to the direction of the non-uniform electric field E" id="MathJax-Element-765-Frame" role="presentation" style="position: relative;" tabindex="0">EE $\mathbf{E}$ , the force F" id="MathJax-Element-766-Frame" role="presentation" style="position: relative;" tabindex="0">FF $\mathbf{F}$ and torque τ" id="MathJax-Element-767-Frame" role="presentation" style="position: relative;" tabindex="0">ττ $\tau$ on the dipole can be determined as follows:

The force F" id="MathJax-Element-768-Frame" role="presentation" style="position: relative;" tabindex="0">FF $\mathbf{F}$ on a dipole in a non-uniform electric field is given by:

\(\mathbf{F} = (\mathbf{p} \cdot \nabla) \mathbf{E}\)

Since the field is non-uniform, F" id="MathJax-Element-769-Frame" role="presentation" style="position: relative;" tabindex="0">FF $\mathbf{F}$ is not zero.

The torque τ" id="MathJax-Element-770-Frame" role="presentation" style="position: relative;" tabindex="0">ττ $\tau$ on the dipole is given by:

\(\tau = \mathbf{p} \times \mathbf{E}\)

When p" id="MathJax-Element-771-Frame" role="presentation" style="position: relative;" tabindex="0">pp $\mathbf{p}$ is parallel to E" id="MathJax-Element-772-Frame" role="presentation" style="position: relative;" tabindex="0">EE $\mathbf{E}$ , the cross product is zero, hence τ=0" id="MathJax-Element-773-Frame" role="presentation" style="position: relative;" tabindex="0">τ=0τ=0 $\tau = 0$ .

Therefore, for an electric dipole in a non-uniform electric field with the dipole moment parallel to the direction of the field, the force and torque on the dipole respectively are:

The correct option is (2) F ≠ 0, τ = 0.

CONCEPT:

• Electric dipole: When two equal and opposite charges are placed at a very small distance to each other then this arrangement is called an electric dipole. 
• Electric dipole moment: It is defined as the product of the magnitude of one charge and the distance between the charges in an electric dipole.

⇒ P = q × 2r

Where 2r = distance between the two charges

EXPLANATION:

• The electric dipole moment is defined as the product of the magnitude of one charge and the distance between the charges in an electric dipole.

⇒ P = q × 2r     -----(1)

Where P = electric dipole moment, 2r = = distance between the two charges and q = charge

• As we know, the SI unit of charge is Coulomb and that of distance is meter.
• Therefore, the SI unit of electric dipole moment is Coulomb-metre.
• Hence, option 3 is correct.

CONCEPT:

• Electric dipole: When two equal and opposite charges are separated by a small distance then this combination of charges is called an electric dipole.
• Electric dipole moment: The multiplication of charge and the distance between them is called an electric dipole moment.
  • The electric dipole moment is denoted by P.
  • The SI unit of dipole moment is Coulombmeter (Cm)

The Strength of Dipole moment = \(\vec P\) = q × d

Where q is charge and d is the distance between two charged particles.

CALCULATION:

Given that:

The magnitude of Charge of each particle (q) = 20 C

Distance between them (d) = 2 cm = 2 × 10^-2 m 

Electric dipole moment (P) = q × d = 20 C × 2 × 10-2 m = 0.4 C.m

Hence option 3 is correct.

The torque for a dipole placed in the electric field is given by:

\(\vec \tau = \vec p \times \vec E\)

Since it is a cross product, we conclude that the torque exerted by the field on the dipole is perpendicular to both the field and the dipole moment

Derivation:

The measure of force that causes an object to rotate about an axis is known as torque. Torque is a vector quantity and its direction depends on the direction of the force on the axis. The magnitude of the torque vector is calculated as follows:

τ = Fr sinθ

where r is the length of the moment arm

θ is the angle between the moment arm and the force vector

Electric Dipole: A pair of electric charges with an equal magnitude but opposite charges separated by a distance d is known as an electric dipole.

The electric dipole moment is a vector having a defined direction from the negative charge to the positive charge.

\(\vec p = q\vec d\)

Consider a dipole with charges +q and –q forming a dipole since they are a distance d away from each other. Let it be placed in a uniform electric field of strength E such that the axis of the dipole forms an angle θ with the electric field.

The force on the charges is:

\({\vec F_ + } = + q\vec E\)

\({\vec F_ - } = - q\vec E\)

The components of force perpendicular to the dipole are:

\(F_ + ^ + = + qE\sin \theta \)

\(F_ - ^ + = - qE\sin \theta \)

Since the force magnitudes are equal and are separated by a distance d, the torque on the dipole is given by:

Torque (τ) = Force × distance separating forces

τ = d qE sin θ

Since dipole moment is given by p = qd, and the direction of the dipole moment is from the positive to the negative charge; it can be seen from the above equation that the torque is the cross product of the dipole moment and electric field.

Notice that the torque is in the clockwise direction (hence negative) in the above figure if the direction of the Electric field is positive.

∴ τ = -pE sin θ, or

\(\vec \tau = \vec p \times \vec E\)

\(\left| {\vec \tau } \right| = \left| {\vec p \times \vec E} \right| = pE\sin \theta \)

Concept:

• Assume an electric dipole is placed in a uniform electric field as shown in figure.
• Each charge of dipole experiences a force qE in electric field.
• Since points of action of these forces are different, these equal and anti-parallel forces give rise to a couple that rotate the dipole and make the dipole to align in the direction of field.
• The torque τ experienced by the dipole is (qE) × (2dsinθ), where, 2d is the length of dipole and θ is the angle between dipole and field direction.

Explanation:

From the above explanation we can say that an electric dipole generally experiences a force and a torque both

Notes:

• When the dipole is placed in uniform electric field, net force on the dipole is zero.
• When the dipole is placed in non-uniform electric field, it experiences both force as well as torque.

CONCEPT:

• Consider an electric dipole consisting of two equal and opposite point charge -q at A and +q at point B separated by a small distance AB = 2l, having dipole moment p = q x 2a directed from -q to +q along the axis of the dipole.

EXPLANATION:

• Let this dipole be placed in a uniform electric field E at an angle θ with the direction of E. Force on charge +q at A = qE, along the direction of E. Force on charge -q at B = qE, along the direction opposite to E.
• Since electric field (E) is uniform, therefore, net force on the dipole is 0. However, as the forces are equal, unlike and parallel, acting at different points, therefore, they form a couple which rotates the dipole. Hence, the couple tends to align the dipole along the direction of the electric field (E).

∴ τ = force × arms of couple

τ = F × 2asinθ = (qE) × 2asinθ

τ = (q × 2a)E sinθ             \(\left| {\vec p} \right| = \left( {q\;x\;2a} \right)\)

τ = pE sinθ

\(\vec \tau = \vec p \times \;\vec E\)

CONCEPT:

• Electric dipole: When two equal and opposite charges are placed at a very small distance to each other then this arrangement is called an electric dipole. 
• Electric dipole moment: It is defined as the product of the magnitude of one charge and the distance between the charges in an electric dipole.

⇒ P = q × 2r     -----(3)

Where 2r = distance between the two charges

CALCULATION:

Given q = 3.2 × 10−19 coulomb, 2r = 2.4 Å = 2.4 × 10^-10 m and E = 4 × 105 volt/m

• The dipole moment does not depend on the electric field, therefore the dipole moment of a dipole is

⇒ P = q × 2r

⇒ P = 3.2 × 10−19 × 2.4 × 10-10

⇒ P = 7.68 × 10^-29 C-m

• Hence, option 3 is correct.

CONCEPT:

• An electric dipole is a system of two equal and opposite charges separated by a fixed distance and the strength of a dipole is measured by dipole moment. It is calculated as P = q x 2a

EXPLANATION

• When a dipole is placed in an electric field each charge experiences a force that is equal to F = qE.
• Force on a negative charge is F_-q = -qE and its direction are in the opposite of direction of the electric field.
• Force on positive charge q is F_q = qE and its direction is in the direction of the electric field.

• The net force on the dipole is F = F_-q+ F_q = -qE + qE = 0
• So, the correct answer will be option 1.

Important Points 

• When the dipole is placed in a uniform electric field, the net force on the dipole is zero.
• When the dipole is placed in a non-uniform electric field, it experiences both forces as well as torque.

CONCEPT:

The potential energy of electric dipole in an external electric field is written as

\(U = - \overrightarrow P .\overrightarrow E \) -----(1)

Where P is the dipole and E is the electric field.

CALCULATION:

Using equation (1) we get;

\(U = - \overrightarrow P .\overrightarrow E \)

⇒ U = – PEcosθ 

The angle between the electric field and the electric dipole is 180°, therefore,

U = –PEcos180°

U = + PE

On moving towards the right electric field strength decrease therefore potential energy decrease.

The net force on the electric dipole is towards the right and the net torque acting on it is zero.

So, it will move towards the right.

Given:

A short dipole in air or vacuum

Concept:

An electric dipole is is a system of two-point charges of equal and opposite magnitude placed at a short distance.

Formula:

For a dipole, dipole moment,

p = q(2l)

Calculations:

E(+q) = \(\frac{q}{4\pi \epsilon 0(r-l)^2}\) towards BP

E(-q) = \(\frac{q}{4\pi \epsilon 0(r+l)^2}\) towards PA

E = E(+q) - E(-q)

⇒ E = \(\frac{q}{4\pi \epsilon 0(r-l)^2} -\frac{q}{4\pi \epsilon 0(r+l)^2}\)

⇒ E = \(\frac{4qlr}{4\pi \epsilon 0(r^2-l^2)^2}\)

⇒ E = \(\frac{2q(2l)r}{4\pi \epsilon 0(r^2-l^2)^2}\)

But q(2l) = p and for a short dipole r² - l² ≈ r² as l << r

⇒ E = \(\frac{2pr}{4\pi \epsilon 0r^4}\)

⇒ E= \(\frac{2p}{4\pi \epsilon 0r^3}\)

CONCEPT:

• Electric dipole: When two equal and opposite charges are separated by a small distance then this combination of charges are called as electric dipole.
• The multiplication of charge and the distance between them is called as electric dipole moment.
• The electric dipole moment is denoted by P and the SI unit of dipole moment is Coulombmeter (Cm)

Dipole moment = P = q × d

Where q is charge and d is distance between two charge particles.

• The direction electric dipole moment is from negative charge to positive charge.
• The space or region around the electric charge in which electrostatic force can be experienced by other charge particle is called as electric field by that electric charge.

The electric field on axial line (at point B):

\(\overrightarrow {{E_{ax}}} = \frac{1}{{4\pi {_0}}}\frac{{2\;\vec P}}{{{r^3}}}\)

The electric field on equatorial line (at point A):

\(\overrightarrow{{{E}_{eq}}}=\frac{-~1}{4\pi {{\epsilon }_{0}}}\frac{~\vec{P}}{{{r}^{3}}}\)

Where  is dipole moment of the electric dipole, ϵ₀ is permittivity of free space and r is distance of points A and B from the centre of the dipole.

EXPLANATION:

According to the given formula of the electric field at two points A and B:

\(\overrightarrow{{{E}_{ax}}}=\frac{1}{4\pi {{\epsilon }_{0}}}\frac{2~\vec{P}}{{{r}^{3}}}\)

\(\overrightarrow{{{E}_{eq}}}=\frac{-~1}{4\pi {{\epsilon }_{0}}}\frac{~\vec{P}}{{{r}^{3}}}\)

\(\overrightarrow{{{E}_{ax}}}=-2~\overrightarrow{{{E}_{eq}}}\) So option 3 is correct.

• The direction of electric field on axial line is along in the direction of dipole moment.
• The direction of electric field on equatorial line is along in the direction opposite to that of dipole moment.