What is the new capacitance of the plates, when a slab of Dielectric Constant K and thickness one-fourth of the separation of plate is inserted between the plates?a)4KCo/K + 2b)4KCo/1 + 2Kc)4KCo/K + 3d)4KCo/1 + 3KCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

As the slab is inserted between the plates,
V = E₀(d/4)/K + E_o(3d/4)
= E_od (1/4K + 3/4)
= V_o(1+3K)/4K
Now we know, C = Q_o/V
= 4KC_o/(1 + 3K)

Answer

As the slab is inserted between the plates,
V = E₀(d/4)/K + E_o(3d/4)
= E_od (1/4K + 3/4)
= V_o(1+3K)/4K
Now we know, C = Q_o/V
= 4KC_o/(1 + 3K)

Answer

Explanation:

When a dielectric material is inserted between the plates of a capacitor, it affects the capacitance of the capacitor. The dielectric material has a property called the dielectric constant (K), which determines how much the capacitance changes.

The formula for the capacitance of a parallel plate capacitor is given by:

C = ε0 * A / d

Where:
C is the capacitance
ε0 is the permittivity of free space (8.85 x 10^-12 F/m)
A is the area of the plates
d is the separation between the plates

Effect of Dielectric:
When a dielectric material is inserted between the plates, the capacitance increases. The dielectric constant (K) determines the extent of this increase. The formula for the new capacitance with a dielectric is given by:

C' = K * C

Where:
C' is the new capacitance with the dielectric
K is the dielectric constant
C is the original capacitance without the dielectric

In this case, the thickness of the dielectric material is one-fourth of the separation between the plates. Let's denote the separation between the plates as 'd' and the thickness of the dielectric as 't'. According to the given information, t = d/4.

Calculating the new capacitance:
To calculate the new capacitance, we need to find the original capacitance without the dielectric (C) and then apply the formula C' = K * C.

The original capacitance without the dielectric can be calculated using the formula:

C = ε0 * A / d

Since the area of the plates (A) is not given, we can assume it to be 1 (unit area) for simplicity. Plugging in the values, we get:

C = ε0 / d

Now, we can calculate the new capacitance (C'):

C' = K * C
= K * (ε0 / d)
= K * ε0 / d

Since the thickness of the dielectric (t) is one-fourth of the separation between the plates (d), we can substitute t = d/4:

C' = K * ε0 / (d/4)
= 4K * ε0 / d

The permittivity of free space (ε0) is a constant value, so we can combine it with the constant 4K:

C' = 4K * ε0 / d

Therefore, the new capacitance with the dielectric is 4K * ε0 / d, which corresponds to option 'D': 4KCo/1 3K.

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