SP212 Lesson 11Ch. 25, 1-3 – Capacitance

January 3, 2020

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Capacitors

We’ve seen capacitors in lab. In essence, a capacitor consists of two isolatedconductors of area A, separated by a distance d . These conductors are calledplates, regardless of their shape and we say the capacitor is charged when oneplate has charge +q and the other −q.

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The equal but opposite charges on a conductor create a potential difference Vbetween the two plates. This related to the charge q:

q = CV

The SI unit of capacitance is the farad (F) (1 F = 1 C/V).

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Capacitors in circuits

As you know, capacitors are mainly used in circuits. They can “charge” or“discharge.”:

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Calculating the Capacitance

We need to be able to calculate the capacitance of capacitors with variousgeometries. We will follow three basic steps:

Use Gauss’s Law to relate the charge q on the positive plate and the electricfield ~E

Integrate along the field (from negative to positive) to find the potential

Plug both of these into q = CV

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The Parallel Plate Capacitor

We draw our Gaussian surface to just enclose the charge on the positive plate:qenc = qOur clever choice of Gaussian surface also makes the flux easy:

Φ =

∮~E · d~A

= EA

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The Parallel Plate Capacitor

q = ε0EA

We need to find V by integrating the field, which we do from the negative topositive plate:

V =

∫ +

−E ds = E

∫ d

0

ds = Ed

Plugging q and V into q = CV , we solve for C for a parallel pate capacitor:

C =ε0A

d

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Other Capacitors

Cylindrical: C = 2πε0L

ln( ba )

Spherical: C = 4πε0abb−a

Isolated Sphere: C = 4πε0R

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Capacitors in Circuits

When electrical components (such as capacitors) are connected into circuits, weoften want to simplify the circuit by finding an equivalent capacitance for all (orat least most) of them.

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Series vs. Parallel

Capacitors in Parallel

Capacitors in Series

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Finding the Equivalent Capacitance

Capacitors in parallel add like you’re used to:

Ceq =n∑

i=1

Ci

Capacitors in series add “weird” (that’s a technical term):

1

Ceq=

n∑i=1

1

Ci

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Challenge: Series vs. Parallel

In (a) are capacitors 1 and 3 in series?

In (b) are capacitors 1 and 2 in parallel?

Rank the three circuits by their equivalent capacitance, greatest first.

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Challenge: Equipotentials

The figure shows three circuits, each containing a switch and two capacitors thatare initially charged as shown in the figure. After the switches have been closed,in what circuit (if any) will the charge on the left hand capacitor....

(a) increase

(b) decrease

(c) remain the same

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