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Chapter 17: Problem 25

A parallel plate capacitor is charged. If the plates are pulled apart (a) the capacitance increases (b) the potential difference increases (c) the total charge increases (d) the charge and potential difference remain the same

Short Answer

Expert verified
(b) the potential difference increases.

Step by step solution

01

Understanding Parallel Plate Capacitors

A parallel plate capacitor consists of two conductive plates separated by a distance. When charged, it holds an electric charge and electrical energy.
02

Formula for Capacitance of a Parallel Plate Capacitor

The capacitance of a parallel plate capacitor is given by the formula: \[ C = \frac{\varepsilon_0 \cdot A}{d} \] where \( C \) is the capacitance, \( \varepsilon_0 \) is the permittivity of free space, \( A \) is the area of one of the plates, and \( d \) is the distance between the plates.
03

Effect of Increasing Distance on Capacitance

If the distance \( d \) between the plates is increased, according to the formula for capacitance, \( C \) decreases because they are inversely proportional.
04

Revisiting Potential Difference Formula

The potential difference \( V \) across the plates is given by the formula: \[ V = \frac{Q}{C} \] where \( Q \) is the charge and \( C \) is the capacitance. Since \( C \) decreases when \( d \) is increased, \( V \) must increase if \( Q \) remains constant.
05

Analyzing Charge Conservation

The total charge \( Q \) on the capacitor remains constant when the plates are pulled apart, as there is no external circuit allowing charge to enter or leave the capacitor.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Capacitance
Capacitance is a measure of how much electric charge a conductor can store per unit of potential difference. For a parallel plate capacitor, it specifically depends on the physical characteristics of the capacitor itself, such as the area of the plates and the separation distance. Using the formula
  • \( C = \frac{\varepsilon_0 \cdot A}{d} \)
  • where \( C \) is capacitance, \( \varepsilon_0 \) is the permittivity of free space, \( A \) is the plate area, \( d \) is the separation distance.
One key aspect to remember is that capacitance is inversely proportional to the distance between plates. This means that as the plates are moved farther apart, the capacitance of the capacitor decreases because the electric field strength between the plates is diminished. This is important because it affects how much charge the capacitor can store for a given potential difference.
Potential Difference
The potential difference, often referred to as voltage, is the energy per unit charge needed to move a charge between two points in an electric field. For a capacitor, the potential difference across its plates is derived from the relationship:
  • \( V = \frac{Q}{C} \)
  • where \( V \) is the potential difference, \( Q \) is the charge, and \( C \) is the capacitance.
When the distance between the capacitor's plates increases, the charge \( Q \) remains constant as long as the capacitor is isolated from any external circuit. However, the capacitance \( C \) decreases, causing the potential difference \( V \) to increase. Think of this as needing more work to move the same amount of charge, due to a weaker electric field when the plates are farther apart.
Electric Charge
Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. In a parallel plate capacitor:
  • The charge \( Q \) stored is directly related to the capacitance \( C \), and potential difference \( V \)
  • \( Q = C \times V \)
When the plates of a capacitor are pulled apart, without adding or removing charge, the total charge remains the same. This is an important conservation principle, meaning that in a closed system (no external circuit), the charge doesn't spontaneously increase or decrease. Charging a capacitor usually involves connecting it to a power source which then populates the plates with opposite charges until equilibrium (balance) is reached, reflecting the amount the capacitor can store at a given voltage.
Permittivity of Free Space
Permittivity of free space, denoted \( \varepsilon_0 \), is a fundamental physical constant that quantifies how an electric field affects, and is affected by, a vacuum. In the context of a parallel plate capacitor:
  • It determines how the capacitance \( C \) depends on the area \( A \) and distance \( d \) between the plates.
  • The equation \( C = \frac{\varepsilon_0 \cdot A}{d} \) showcases its role.
Understanding \( \varepsilon_0 \) helps to realize that it is a measure of the ability of a vacuum to permit electric field lines. The higher the permittivity, the greater the ability to store electrical energy within the capacitor's electric field. Essentially, \( \varepsilon_0 \) is like a bridge that governs how well the electric field operates in a given space, affecting the interaction between the electric charges across the plates.

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Most popular questions from this chapter

The capacities of three capacitors are in the ratio \(1: 2: 3\). Their equivalent capacity when connected in parallel is \(\frac{60}{11} \mu \mathrm{F}\) more than that when they are connected in series. The individual capacitors (in \(\mu \mathrm{F}\) ) are of capacities (a) \(4,6,7\) (b) \(1,2,3\) (c) \(2,3,4\) (d) \(1,3,6\)

The capacitance \(C\) of a capacitor is (a) independent of the charge and potential of the capacitor (b) dependent on the charge and independent of potential (c) independent of the geometrical configuration of the capacitor (d) independent of the dielectric medium between the two conducting surfaces of the capacitor

Two capacitors \(C_{1}\) and \(C_{2}=2 C_{1}\) are connected in a circuit with a switch between them as shown in the figure. Initially the switch is open and \(C_{1}\) holds charge \(Q\). The switch is closed. At steady state, the charge on each capacitor will be (a) \(Q, 2 Q\) (b) \(\frac{Q}{3}, \frac{2 Q}{3}\) (c) \(\frac{3 Q}{3}, 3 Q\) (d) \(\frac{2 Q}{3}, \frac{4 Q}{3}\)

A capacitor of capacitance \(C\) is charged to a constant potential difference \(V\) and then connected in series with an open key and a pure resistor \(R\). At time \(t=0\), the key is closed. If \(I\) is the current at time \(t=0\), a plot of \(\log I\) against \(t\) is shown as (1) in the graph. Later one of the parameters, i.e., \(V, R\) and \(C\) is changed, keeping the other two constant and graph ( 2 ) is recorded. Then (a) \(C\) is reduced (b) \(C\) is increased (c) \(R\) is reduced (d) \(R\) is increased

The plates of a capacitor are charged to a potential difference of 320 volt and are then connected to a resistor. The potential difference across the capacitor decays exponentially with time. After 1 sec, the potential difference between the plates of the capacitor is \(240 \mathrm{~V}\), then after 2 and 3 seconds the potential difference between the plates will be, respectively (a) \(200 \mathrm{~V}\) and \(180 \mathrm{~V}\) (b) \(180 \mathrm{~V}\) and \(135 \mathrm{~V}\) (c) \(160 \mathrm{~V}\) and \(80 \mathrm{~V}\) (d) \(140 \mathrm{~V}\) and \(20 \mathrm{~V}\)
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