Question

What is the radius of an isolated spherical conductor that has a capacitance of \(1.00 \mathrm{~F} ?\)

Short answer

Answer: The radius of the isolated spherical conductor is approximately \(8.92 \times 10^6 \mathrm{m}\).

Step-by-step solution

Recall the Formula for the Capacitance of a Spherical Conductor

The formula for capacitance (C) of an isolated spherical conductor is given by: C = \(4\pi\epsilon_0 R\) Where \(\epsilon_0\) is the permittivity of free space (\(8.854 \times 10^{-12} \mathrm{F/m}\)) and R represents the radius of the sphere.

Rearrange the Formula to Solve for the Radius

In order to find the radius (R), we will rearrange the formula above to have R by itself on one side: R = \(\frac{C}{4\pi\epsilon_0}\)

Substitute the given Capacitance Value and Calculate the Radius

The given capacitance value is \(1.00 \mathrm{F}\), so now we will substitute this value into the formula to find the radius: R = \(\frac{1.00 \mathrm{F}}{4\pi(8.854 \times 10^{-12} \mathrm{F/m})}\) Now, we will perform the calculation to find the radius. R = \(\frac{1.00}{4\pi(8.854 \times 10^{-12})} \approx 8.92 \times 10^6 \mathrm{m}\) Thus, the radius of the isolated spherical conductor is approximately \(8.92 \times 10^6 \mathrm{m}\).

Key concepts

Permittivity of Free Space

The permittivity of free space, also known as the vacuum permittivity, is a fundamental constant in physics. It represents the ability of free space to permit electric field lines to pass through. Often denoted by the symbol \(\varepsilon_0\), this constant is crucial when dealing with equations involving electromagnetic fields.
This value is extremely small and is approximately \(8.854 \times 10^{-12} \, \mathrm{F/m}\). The unit here, \(\mathrm{F/m}\) (farads per meter), relates to the capacitance per unit length of a medium. Permittivity is a measure of how much resistance is encountered when forming an electric field in a vacuum.
Understanding this constant is essential when calculating the capacitance of systems such as spherical conductors. It shows how electric fields interact with their surroundings, helping to determine other important properties like capacitance and radius.

Capacitance Formula

The capacitance formula for a spherical conductor is something every student dealing with electrostatics should be familiar with. Represented by \(C = 4\pi\varepsilon_0 R\), this formula allows you to compute the capacitance \(C\) of an isolated spherical conductor.
Here's a breakdown of the formula:

• \(C\) indicates the capacitance, a measure of the conductor's ability to store electrical charge.
• \(4\pi\) is a constant that arises from the geometry of a sphere, reflecting the symmetrical nature of the field around it.
• \(\varepsilon_0\), the permittivity of free space, describes how the electric field interacts and propagates through space.
• \(R\), the radius of the sphere, is what we're often solving for when designing or analyzing spherical capacitors.

Using this equation, you can not only find the capacitance but also rearrange it to solve for the radius of the conductor when the capacitance is known. This flexibility makes it an invaluable tool in physics.

Radius Calculation

Calculating the radius of a spherical conductor is a straightforward application of the capacitance formula, specifically when the capacitance is known. To draw the radius \(R\) from the formula, we need to rearrange the original capacitance equation like this:
\[ R = \frac{C}{4\pi\varepsilon_0} \]
This rearranged formula shows how capacitance and permittivity determine the geometric property of the sphere.
When given a specific capacitance, like \(1.00 \, \mathrm{F}\), substituting into the formula becomes an excellent practice problem for understanding spherical capacitors. By inputting our known values, including substituting for \(\varepsilon_0\) with \(8.854 \times 10^{-12} \, \mathrm{F/m}\), you can solve for \(R\) to determine the physical size of your conductor.
This ability to compute the radius is not only a theoretical exercise but also practical when designing real-world circuits. The understanding here extends beyond memorization to appreciation of how components function in various environments.