What is the capacitance of a pair of circular plates with a radius of 4.0  cm  separated by 3.2  mm  of mica? The dielectric constant of mica is 7.
Express your answer using one significant figure and include the appropriate units.

The capacitance \( C \) of a parallel-plate capacitor is given by the formula:

\[
C = \frac{\varepsilon_0 \varepsilon_r A}{d}
\]

where:
- \( \varepsilon_0 \) is the permittivity of free space \( (8.85 \times 10^{-12} \, \text{F/m}) \),
- \( \varepsilon_r \) is the relative permittivity (dielectric constant),
- \( A \) is the area of one of the plates \( A = \pi r^2 \),
- \( d \) is the separation between the plates.

### Given values:
- \( r = 4.0 \, \text{cm} = 0.04 \, \text{m} \),
- \( d = 3.2 \, \text{mm} = 0.0032 \, \text{m} \),
- \( \varepsilon_r = 7 \) (dielectric constant of mica).

Now, let's calculate the capacitance.

First, the area of the plates:
\[
A = \pi r^2 = \pi (0.04)^2 \, \text{m}^2
\]

Next, we plug all values into the capacitance formula to find \( C \).

\[
C = \frac{(8.85 \times 10^{-12})(7)(\pi (0.04)^2)}{0.0032}
\]

Let's compute that.The capacitance of the pair of circular plates is approximately \( 9.7 \times 10^{-11} \, \text{F} \), or \( 97 \, \text{pF} \) (picofarads) when rounded to one significant figure.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does increasing the plate separation affect the capacitance?
2. What happens to the capacitance if the radius of the plates is doubled?
3. How would the capacitance change if the dielectric constant were higher?
4. What is the significance of the dielectric material in a capacitor?
5. How do we calculate energy stored in a capacitor with this capacitance?

**Tip:** Dielectric materials increase capacitance by reducing the electric field between the plates, effectively allowing more charge to be stored.

What is the capacitance of a pair of circular plates with a radius of 4.0 cm separated by 3.2 mm of mica? The dielectric constant of mica is 7. Express your answer using one significant figure and include the appropriate units.

Calculate the capacitance of a pair of circular plates with a radius of 4.0 cm, separated by 3.2 mm of mica with a dielectric constant of 7. This solution covers the use of the parallel-plate capacitor formula and geometric calculations for the plate area. Suitable for college-level physics students.

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