Effective Capacitance with Parallel Dielectrics and Capacitor

❓ Question

Space between the plates of a parallel plate capacitor of plate area:

$$4\text{ cm}^2$$

and separation:

$$d=1.77\text{ mm}$$

is filled with dielectric material having dielectric constant:

$$K=3$$

as shown in figure.

Another capacitor of capacitance:

$$7.5\text{ pF}$$

is connected in parallel to it.

The effective capacitance of the combination is ______ pF.

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🖼 Question Image

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✍️ Short Explanation 

 This problem is based on: 

 👉 Parallel plate capacitor 

 👉 Dielectric medium 

 👉 Parallel combination of capacitors. 

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🔷 Step 1 — Convert SI Units 💯

Plate area:

$$A=4\text{ cm}^2$$
$$=4\times10^{-4}\text{ m}^2$$

Plate separation:

$$d=1.77\text{ mm}$$
$$=1.77\times10^{-3}\text{ m}$$

Dielectric constant:

$$K=3$$

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🔷 Step 2 — Find Capacitance of First Capacitor

Using:

$$C=\frac{K\varepsilon_0A}{d}$$

Substitute values:

$$C= \frac{ 3(8.85\times10^{-12})(4\times10^{-4}) }{ 1.77\times10^{-3} }$$

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🔷 Step 3 — Simplify

Numerator:

$$3\times8.85\times4 = 106.2$$

So:

$$C= \frac{ 106.2\times10^{-16} }{ 1.77\times10^{-3} }$$
$$= 60\times10^{-13}$$
$$=6\times10^{-12}\text{ F}$$
$$=6\text{ pF}$$

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🔷 Step 4 — Parallel Combination

Another capacitor:

$$7.5\text{ pF}$$

In parallel:

$$C_{eq}=6+7.5$$
$$=13.5\text{ pF}$$

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🔷 Step 5 — JEE Trap Alert 🚨

❌ cm$^2$ to m$^2$ conversion bhool jaana

❌ mm to m conversion galat kar dena

❌ Parallel me reciprocal formula use kar lena

Remember:

For parallel connection:

$$\boxed{ C_{eq}=C_1+C_2 }$$

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✅ Final Answer

$$\boxed{13.5\text{ pF}}$$

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