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Capacitance of a capacitor becomes 7/6 times its original value if a dielectric slab of thickness, t = 2/3 d is introduced in between the plates. 'd' is the separation between the plates. The dielectric constant of the dielectric slab is
  • a)
    14/11
  • b)
    11/14
  • c)
    7/11
  • d)
    11/7
Correct answer is option 'A'. Can you explain this answer?
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Given:
- Capacitance of the capacitor becomes 7/6 times its original value when a dielectric slab of thickness t = 2/3 d is introduced between the plates.
- d is the separation between the plates.

To find:
The dielectric constant of the dielectric slab.

Let's solve this step by step.

Step 1: Understanding the concept
The capacitance of a parallel plate capacitor is given by the formula:

C = ε₀A/d

where C is the capacitance, ε₀ is the permittivity of free space, A is the area of the plates, and d is the separation between the plates.

When a dielectric material is introduced between the plates, the capacitance increases by a factor of κ, which is the dielectric constant of the material. The new capacitance is given by:

C' = κε₀A/d

Step 2: Calculating the original capacitance
Let's assume the original capacitance is C₀.

C₀ = ε₀A/d

Step 3: Calculating the new capacitance
The capacitance becomes 7/6 times its original value, so the new capacitance is:

C' = (7/6)C₀

C' = (7/6)ε₀A/d

Step 4: Introducing the dielectric slab
When the dielectric slab is introduced, the thickness t = 2/3 d.

Therefore, the new separation between the plates is:

d' = d - t

Substituting the values in the formula for C':

C' = (7/6)ε₀A/d'

C' = (7/6)ε₀A/(d - t)

Step 5: Calculating the dielectric constant
We need to find the dielectric constant κ.

C' = κε₀A/d'

(7/6)C₀ = κε₀A/(d - t)

C₀ = κε₀A/d

Dividing both equations:

(7/6)C₀ / C₀ = κε₀A/(d - t) / (κε₀A/d)

7/6 = (d - t)/d

7d/6 = d - t

7d = 6d - 6t

6t = d

Substituting t = 2/3 d:

6(2/3 d) = d

4d = d

d = 4

Therefore, the separation between the plates is 4.

Step 6: Calculating the dielectric constant (contd.)
Substituting d = 4 in the equation:

6t = d

6t = 4

t = 4/6

t = 2/3

Therefore, the thickness of the dielectric slab is 2/3.

Step 7: Calculating the dielectric constant (contd.)
Substituting t = 2/3 and d = 4 in the equation:

7d/6 = d - t

7(4)/6 = 4 - 2/3

28/6 = 12/3 - 2/3

28/6 = 10/3

Cross-multiplying:

3(28) = 6(10)

84 = 60

This is a contradiction, so our assumption is incorrect.

Therefore, the dielectric constant of the
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Capacitance of a capacitor becomes 7/6 times its original value if a dielectric slab of thickness, t = 2/3 d is introduced in between the plates. 'd' is the separation between the plates. The dielectric constant of the dielectric slab isa)14/11b)11/14c)7/11d)11/7Correct answer is option 'A'. Can you explain this answer?
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