Capacitance (in F) of a spherical conductor having radius 1 m, is
(a) 1.1 x 10^{-10}
(b) 10^{-6}
(c)

Q: Why does a single conductor have capacitance?

Capacitance is the ability to store charge per unit potential. Even a single conductor can store charge; its 'other plate' is considered to be at infinity.

Q: What is the value of $4\pi\varepsilon_0$?

Since $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9$, $4\pi\varepsilon_0$ is the reciprocal, $\frac{1}{9 \times 10^9}$.

Q: Why is the capacitance so small?

1 Farad is a massive unit of capacitance. Even a sphere the size of the Earth has a capacitance of only about $711 \mu\text{F}$. A 1-meter sphere will naturally have a very small capacitance in the picofarad range.

Q: How do we convert $0.11 \times 10^{-9}$ to $1.1 \times 10^{-10}$?

Multiply the decimal part by 10 and divide the exponent part by 10 (which means subtracting 1 from the exponent): $0.11 \times 10 = 1.1$ and $10^{-9} / 10 = 10^{-10}$.

Q: Does the material of the spherical conductor matter?

No, the capacitance of an isolated conductor depends only on its size and shape (geometry), and the medium surrounding it, not on the material it is made of.

Question

Capacitance (in F) of a spherical conductor having radius $1	ext{ m}$, is

Options:

(A) $1.1 	imes 10^{-10}$

(B) $10^{-6}$

(C) $9 	imes 10^{-9}$

(D) $10^{-3}$

Sigma Percentile · Accepted Answer

### The Concept of an Isolated Capacitor

When we hear the word "capacitor," we usually picture two parallel plates separated by some distance. But did you know that a single, isolated conductor can also act as a capacitor?

Imagine a perfectly spherical conductor floating in the vast emptiness of free space. Even though there isn't a second plate nearby, this sphere can still store electric charge. In physics, we consider the "second plate" of this isolated capacitor to be located at infinity, where the electric potential is zero.

### The Master Equation

The capacitance of any object tells us how much charge it can store for a given rise in potential. For an isolated spherical conductor of radius $R$, the capacitance $C$ is given by a beautifully simple formula:

$$C = 4\pi\varepsilon_0 R$$

Notice something fascinating here? The capacitance depends **only** on the geometry of the conductor (its radius $R$) and the medium surrounding it (represented by the permittivity $\varepsilon_0$). It doesn't matter if the sphere is made of copper, gold, or aluminum; its ability to store charge remains exactly the same!

### Executing the Calculation

In our problem, we are given a spherical conductor with a radius $R = 1	ext{ m}$. Let's plug this into our formula.

We know the value of the electrostatic constant:
$$\frac{1}{4\pi\varepsilon_0} = 9 	imes 10^9	ext{ N}\cdot	ext{m}^2/	ext{C}^2$$

Therefore, the term $4\pi\varepsilon_0$ is simply the reciprocal:
$$4\pi\varepsilon_0 = \frac{1}{9 	imes 10^9}$$

Now, substituting $R = 1	ext{ m}$:
$$C = \frac{1}{9 	imes 10^9} 	imes 1$$

Let's compute this value. Dividing $1$ by $9$ gives approximately $0.111$. Bringing the $10^9$ from the denominator to the numerator changes the sign of the exponent:
$$C \approx 0.111 	imes 10^{-9}	ext{ F}$$

### The Final Touch

To match our answer with the given options, we need to adjust the decimal point. By shifting the decimal one place to the right, we must decrease the exponent by one:
$$C = 1.11 	imes 10^{-10}	ext{ F}$$

This perfectly matches option (a).

**Food for thought:** A capacitance of $10^{-10}	ext{ F}$ (or $100	ext{ pF}$) is incredibly small! This shows just how massive the unit of "1 Farad" truly is. To have a capacitance of 1 Farad, an isolated sphere would need a radius of $9 	imes 10^9	ext{ meters}$—that's roughly 13 times the radius of our Sun!

Animated Solution for Physics - Electrostatics: Capacitance (in F) of a spherical conductor having radius $1 m$ , is

Select Answer:

Visualized Solution

Visualizing the Spherical Conductor

Formula for Capacitance

Substituting the Values

Calculating the Value

Final Answer

Food for Thought

The Sigma Insight: Capacitance and Capacitors

The Concept of an Isolated Capacitor

The Master Equation

Executing the Calculation

The Final Touch

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Animated Solution for Physics - Electrostatics: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Capacitance (in F) of a spherical conductor having radius 1 m, is