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Difficulty: MediumCapacitors and Capacitance

A parallel-plate capacitor with air between its plates has a capacitance of 15 μF15\text{ }\mu\text{F}. If the plate separation is reduced to one-third of its initial value and a dielectric material of relative permittivity εr=4.0\varepsilon_r = 4.0 is completely inserted between the plates, what is the new capacitance of the capacitor?

  1. A
    20 μF20\text{ }\mu\text{F}
  2. B
    45 μF45\text{ }\mu\text{F}
  3. C
    60 μF60\text{ }\mu\text{F}
  4. 180 μF180\text{ }\mu\text{F}Answer

Answer

The new capacitance of the capacitor is 180 μF180\text{ }\mu\text{F}.
The capacitance of a parallel-plate capacitor is given by C=εrε0AdC = \frac{\varepsilon_r \varepsilon_0 A}{d}. Reducing plate separation to one-third increases capacitance by a factor of 3. Adding a dielectric with relative permittivity εr=4.0\varepsilon_r = 4.0 increases capacitance by a factor of 4. Combining both effects increases capacitance by a total factor of 3×4=123 \times 4 = 12, yielding 12×15 μF=180 μF12 \times 15\text{ }\mu\text{F} = 180\text{ }\mu\text{F}.

Step-by-Step Solution

1
Express the initial capacitance C1C_1 using the formula for a parallel-plate air capacitor.
C1=ε0Ad=15 μFC_1 = \frac{\varepsilon_0 A}{d} = 15\text{ }\mu\text{F}
Air has a relative permittivity of 1.
2
Write the formula for the modified capacitance C2C_2 with plate separation d′=d3d' = \frac{d}{3} and dielectric constant εr=4.0\varepsilon_r = 4.0.
C2=εrε0Ad′=4.0ε0Ad3=4.0×3×(ε0Ad)=12C1C_2 = \frac{\varepsilon_r \varepsilon_0 A}{d'} = \frac{4.0 \varepsilon_0 A}{\frac{d}{3}} = 4.0 \times 3 \times \left(\frac{\varepsilon_0 A}{d}\right) = 12 C_1
Reducing distance to d/3d/3 increases capacitance by a factor of 3, and inserting the dielectric increases capacitance by a factor of 4.
3
Calculate the value of the new capacitance.
C2=12×15 μF=180 μFC_2 = 12 \times 15\text{ }\mu\text{F} = 180\text{ }\mu\text{F}
Multiplying the combined scaling factor by the initial capacitance gives the final answer.

Key Concept

Parallel Plate Capacitance and Dielectrics
Estimated Time:1m 30s
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Topics

Capacitors and Capacitance

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