A Dipole in an Electric Field MCQ Quiz in मल्याळम - Objective Question with Answer for A Dipole in an Electric Field - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

The correct answer is zero at each point.

Key Points

• If we move from the surface to the center of a charged metal sphere the electric field will be zero at each point​.
• Excess charges placed on a spherical conductor repel and move until they are evenly distributed, as shown in Figure below.
• The excess charge is forced to the surface until the field inside the conductor is zero.
• Outside the conductor, the field is exactly the same as if the conductor were replaced by a point charge at its center equal to the excess charge.

​Additional Information​

• Properties of a conductor in electrostatic equilibrium:
  • The electric field is zero inside a conductor.
  • Just outside a conductor, the electric field lines are perpendicular to its surface, ending or beginning on charges on the surface.
  • Any excess charge resides entirely on the surface or surfaces of a conductor.
• A conductor allows free charges to move about within it.
• The electrical forces around a conductor will cause free charges to move around inside the conductor until static equilibrium is reached.
• Any excess charge will collect along the surface of a conductor.
• Conductors with sharp corners or points will collect more charge at those points.
• A lightning rod is a conductor with sharply pointed ends that collect excess charge on the building caused by an electrical storm and allow it to dissipate back into the air.
• Electrical storms result when the electrical field of Earth’s surface in certain locations becomes more strongly charged, due to changes in the insulating effect of the air.
• A Faraday cage acts as a shield around an object, preventing electric charge from penetrating inside.

Concept:

Electric Dipole:

• Electric Dipole: A pair of equal and opposite charges separated by a distance. The dipole moment (p) is given by \( p = q \times d \), where q is the charge and d is the separation distance.
• Torque on Dipole: When placed in an electric field (E), the torque (τ) experienced by the dipole is given by \( \tau = p \times E \sin \theta \), where θ is the angle between the dipole moment and the electric field.
• SI Unit: The SI unit of torque is Newton-meter (N·m), and the SI unit of electric field is Newton per Coulomb (N/C).
• Dimensional Formula: The dimensional formula for torque is [M L² T²], and for electric field is [M L T³ A¹].
• Important Formula: \( \tau = pE \sin \theta \) and \( p = q \times d \).

Calculation:

Given,

Electric field intensity, E = 2 × 10⁵ N/C

Torque, τ = 4 N·m

Angle, θ = 30°

Dipole length, d = 2 cm = 0.02 m

We know, T = q × d × E × sin30

⇒ 4 = q × d × E × sin30

⇒ 4 = q × 0.02 × 2 × 10⁵ × (1/2)

q = 2 × 10^-3

q = 2mC

∴ The correct option is 4)
\( 4 = q \times 0.02 \times 2 \times 10^5 \times 0.5 \)

Hence, the magnitude of charge on the dipole is 2 mC.

Ans.(2)

Sol.

Equation for work done in rotating dipole from angle 'X₁' to angle 'X₂'

Work done = P * E (cos X₁ - cos X₂)

Here, cos X₁ = cos0 = 1 (stable)

cos X₂ = cos 180 = -1 (unstable)

So,

Work done = P * E (1--1)

= 2PE

Calculation: 

\(\vec{\tau}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{E}}\)... ( torque on diploe)

\(\overrightarrow{\mathrm{p}}=\mathrm{q} \vec{\ell}\)....... ( dipole moment) 

⇒ \(\overrightarrow{\mathrm{E}}=0.2 \frac{\mathrm{V}}{\mathrm{cm}}=20 \frac{\mathrm{V}}{\mathrm{m}}\)

⇒\(\overrightarrow{\mathrm{p}}=4 \times(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})\)

⇒ \( \overrightarrow{\mathrm{p}} =\)\((4 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \mu \mathrm{C}-\mathrm{m}\)

⇒ \(\vec{\tau}=(4 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \times(20 \hat{\mathrm{i}}) \times 10^{-6} \mathrm{Nm}\)

= \((8 \hat{\mathrm{k}}+8 \hat{\mathrm{j}}) \times 10^{-5}=8 \sqrt{2} \times 10^{-5}\)

⇒ α = 2

∴ the value of α is 2.

Concept: 

\(V=\frac{k P \cos \theta}{r^2}\)

& can also checked dimensionally 

∴ The electrostatic potential due to an electric dipole at a distance ‘r’ varies as \(\frac{1}{r^2}\)

The correct answer is option 4) i.e. PE.

CONCEPT:

• Work done in rotating a dipole in an electric field:
  • ​Consider an electric dipole placed in a uniform electric field E as shown.

• The force due to each of the charges is qE and -qE creating a net force. Since they are separated by a distance, a torque is produced. 

Torque, τ = Force × perpendicular distance between charges = qE × 2l sinθ 

Electric dipole moment is given by: p = q × 2l

⇒ τ = pEsinθ 

Therefore τ = p × E (∵ τ, p, and E are vector quantities)

• Work Done to turn an electric dipole from θ₁ to θ₂ is given by: WD = \(\int^{θ_2}_{θ_1} \tau.dθ= \int^{θ_2}_{θ_1}pEsinθ dθ =[pEcosθ]_{θ_1}^{θ_2}\)

⇒ Work Done = pE [cosθ₁ - cosθ₂]

CALCULATION:

Given that: 

To rotate a dipole by 90^∘ ⇒ θ₁ = 0^∘ and θ₂ = 90∘

Work Done = pE [cosθ₁ - cosθ₂] = pE [cos 0^∘ - cos 90^∘] = pE [1 - 0] = pE 

Concept: 

The torque τ experienced by an electric dipole in a uniform electric field is given by

τ = pESinθ 

For maximum torque, the equation simplifies to:

τ = pE....... ( sinθ = sin 90 = 1)

Calculation:

Given

Dipole moment of the first dipole p₁ = 1.2 × 10-30 C-m

Dipole moment of the second dipole p₂ = 2.4 × 10-30 C-m

Electric field for the first dipole E₁ = 5 × 104 NC-1

Electric field for the first dipole E₂ = 15 × 104 NC-1

Maximum Torque for Each Dipole

Torque on the first dipole

⇒τ₁ = p₁E₁ = (1.2 × 10-30) × ( 5 × 104 ) = 6 × 10^-26 C.m

Torque on the second dipole

⇒τ₂ = p₂E₂ =  (2.4 × 10-30) × ( 15 × 104 ) =  36 × 10^-26 C.m

Ratio of Torques:

\(\frac{\tau_1}{\tau_2} = \frac{6}{36} = \frac{1}{6}\)

∴ The value of x = 6

Concept:

Dipole Moment (p ):
The dipole moment is a vector quantity defined as 
P= q.d

 where 
q is the charge and 
d is the separation vector between the charges.

Potential Energy (U) of a Dipole in an Electric Field:
The potential energy of a dipole in a uniform electric field 
E is given by:
U=− p ⋅ E
The dot product 
U =pEcosθ, where 
θ is the angle between the dipole moment p and the electric field E.

Work Done in Rotating the Dipole:
The work done in rotating the dipole from an initial angle to a final angle 
W = Vf – Vi in the electric field is equal to the change in potential energy:
W=U. ​

Calculation:

Work done in rotating the dipole = V_f – V_i

Now, V_f = –PE cos (180°)

V_i = –PE cos 0°

Therefore, W = V_f – V_i

= (–PE cos 180°) – (–PE cos0°)

= 2PE = 2 × 6 × 10^–6 × 1.5 × 10³ = 18 mJ

∴ The correct answer is (18).

CONCEPT:

Electric dipole in a uniform external field:

• We know that when a charge q is placed in electric field E, it experiences a force F, the force is given as,

⇒ F = qE

• So when an electric dipole is placed in the electric field according to the diagram,
• The force on the +q and the -q charge due to the electric field is given as,

⇒ F = qE

• The net force on the electric dipole will be zero.
• The torque on the electric dipole is given as,

⇒ τ = pE.sinθ

\(⇒ \vec{τ} = \vec{p}\times\vec{E}\)

Where θ = angle between the dipole and the electric field

• This torque will tend to align the dipole with the electric field.

EXPLANATION:

• When an electric dipole is placed in an external non-uniform electric field then it is shown as,

     -----(1)

In the diagram,

⇒ E = electric field intensity

From figure 1 it is clear that the force on the charge -q will be,

⇒ F1 = qE      -----(1)

Similarly from figure 1, it is clear that the force on the charge +q will be,

⇒ F2 = qE      -----(2)

• The force F1 and F2 will act opposite to each other.
• The force on the charge -q will act opposite to the electric field while the force on the charge +q acts in the direction of the electric field.

• Since both, the forces are not collinear so the torque will act on the dipole which tries to rotate the dipole anticlockwise. Hence, option 2 is correct.

CONCEPT:

Electric dipole in a uniform external field:

• We know that when a charge q is placed in electric field E, it experiences a force F, the force is given as,

⇒ F = qE

• So when an electric dipole is placed in the electric field according to the diagram,
• The force on the +q and the -q charge due to the electric field is given as,

⇒ F = qE

• The net force on the electric dipole will be zero.
• The torque on the electric dipole is given as,

⇒ τ = pE.sinθ

\(⇒ \vec{τ} = \vec{p}\times\vec{E}\)

Where θ = angle between the dipole and the electric field

• This torque will tend to align the dipole with the electric field.

EXPLANATION:

• When an electric dipole is placed in an external non-uniform electric field then it is shown as,

     -----(1)

In the diagram,

⇒ E = electric field intensity

From figure 1 it is clear that the force on the charge -q will be,

⇒ F1 = qE      -----(1)

Similarly from figure 1, it is clear that the force on the charge +q will be,

⇒ F2 = qE      -----(2)

• The force F1 and F2 will act opposite to each other.
• The force on the charge -q will act opposite to the electric field while the force on the charge +q acts in the direction of the electric field.

• Since both, the forces are not collinear so the torque will act on the dipole which tries to rotate the dipole anticlockwise. Hence, option 2 is correct.