Fundamentals of Physics - Chapter 25: Capacitance

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Questions and Answers

What happens to the capacitance of a cylindrical capacitor if the radius of the inner cylinder is increased?

  • It remains constant
  • It increases
  • It fluctuates
  • It decreases (correct)

For capacitors connected in series, which statement regarding their charge is true?

  • The charge is zero across all capacitors
  • Each capacitor has the same charge (correct)
  • The total charge is the sum of individual charges
  • The charge varies among the capacitors

What is the effect on capacitance when the radius of the outer spherical shell of a spherical capacitor is increased?

  • It becomes undefined
  • It has no effect
  • It decreases
  • It increases (correct)

What is the characteristic of the potential applied to capacitors that are connected in parallel?

<p>They experience the same potential difference (A)</p> Signup and view all the answers

When one charged capacitor is connected in parallel with uncharged capacitors, what occurs when equilibrium is reached?

<p>Charge redistributes among all connected capacitors (B)</p> Signup and view all the answers

What is the relationship between the charge q, capacitance C, and potential difference V in a capacitor?

<p>q = CV (C)</p> Signup and view all the answers

In a parallel-plate capacitor, what happens to the charges on the plates when the circuit is connected and the switch is closed?

<p>One plate gains negative charge while the other gains positive charge. (A)</p> Signup and view all the answers

Which of the following best describes the electric field in a uniform parallel-plate capacitor?

<p>It is uniform in the central region between the plates. (C)</p> Signup and view all the answers

What does the presence of a battery in a circuit with a capacitor imply about the potential difference?

<p>The potential difference is maintained between the terminals of the battery. (C)</p> Signup and view all the answers

In the schematic diagram of a capacitor circuit, what is the role of the switch?

<p>To connect or disconnect the battery from the capacitor. (B)</p> Signup and view all the answers

What is the relationship between the electric field E and the charge q for a parallel-plate capacitor according to Gauss' law?

<p>$E = \frac{q}{\epsilon_0 A}$ (D)</p> Signup and view all the answers

In the context of a parallel-plate capacitor, what assumption allows us to consider the electric field E as constant between the plates?

<p>The plates are infinitely large. (B)</p> Signup and view all the answers

What expression represents the potential difference V between the plates of a capacitor when integrating the electric field E?

<p>$V = \int E \cdot ds$ (A)</p> Signup and view all the answers

How is capacitance C defined for a parallel-plate capacitor using charge q and potential difference V?

<p>$C = \frac{q}{V}$ (A)</p> Signup and view all the answers

Which factor is NOT considered when calculating the capacitance of a parallel-plate capacitor?

<p>The shape of the plates (B)</p> Signup and view all the answers

What is the formula for the capacitance of a cylindrical capacitor given its length L and inner and outer radii a and b?

<p>C = 2πε₀ (L / ln(b/a)) (D)</p> Signup and view all the answers

Which of the following expressions represents the electric field E inside a cylindrical capacitor?

<p>E = (q / (2πε₀L)) (1/r) (A)</p> Signup and view all the answers

For a spherical capacitor, which of the following formulas is used to determine its capacitance?

<p>C = 4πε₀ (ab / (b - a)) (B)</p> Signup and view all the answers

If the separation of the plates in a parallel-plate capacitor is increased while being charged by the same battery, what happens to the stored charge?

<p>The stored charge decreases. (C)</p> Signup and view all the answers

What is the capacitance of an isolated sphere with radius R?

<p>C = 4πRε₀ (C)</p> Signup and view all the answers

Flashcards

Capacitance (C)

The ability of a capacitor to store electrical energy, measured in Farads (F).

Capacitor

A device that stores electrical energy in an electric field.

Gauss's Law

The total electric flux through a closed surface is proportional to the enclosed electric charge.

Voltage (V)

The potential difference between the plates of a capacitor, measured in Volts (V).

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Capacitance Formula

The ratio of the charge on each plate to the potential difference between the plates.

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Capacitance of a Cylindrical Capacitor

The capacitance of a cylindrical capacitor is directly proportional to the length of the capacitor and inversely proportional to the natural logarithm of the ratio of the outer radius to the inner radius.

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Capacitance of a Spherical Capacitor

The capacitance of a spherical capacitor is directly proportional to the product of the radii of the two spheres and inversely proportional to the difference between the radii.

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Capacitance of an Isolated Sphere

The capacitance of an isolated sphere is directly proportional to the radius of the sphere.

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Potential Difference Across a Capacitor

The potential difference across a capacitor is directly proportional to the charge stored on the capacitor and inversely proportional to the capacitance.

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Electric Field Inside a Capacitor

The electric field inside a capacitor is uniform and directed from the positive plate to the negative plate.

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What is Capacitance (C)?

Capacitance is a measure of a capacitor's ability to store charge. It is defined as the ratio of the charge stored on one plate to the potential difference between the plates. The unit of capacitance is the farad (F).

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Potential Difference (V)

The potential difference between the plates of a capacitor, also known as the voltage across the capacitor. It is the amount of work required to move a unit of charge between the plates.

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Charge (Q)

The charge stored on the plates of a capacitor. The amount of charge stored is directly proportional to the capacitance and the potential difference.

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Charging Circuit

A circuit containing a battery, a switch, an uncharged capacitor, and wires. When the switch is closed, electrons flow from the negative terminal of the battery to the capacitor's negative plate, building up charge and creating a potential difference across the capacitor.

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Parallel Capacitors: Same Potential Difference

Capacitors connected in parallel have the same potential difference, which is also the same as their equivalent capacitor.

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Parallel Capacitors: Total Charge

The total charge stored on parallel capacitors is the sum of the charges stored on each individual capacitor.

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Series Capacitors: Same Charge

When capacitors are connected in series, they all have the same charge, which is also the same as their equivalent capacitor.

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Series Capacitors: Total Potential Difference

The potential difference applied to capacitors in series is equal to the sum of the potential differences across each individual capacitor.

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Simplifying Complex Capacitor Circuits

For a circuit with capacitors in parallel and series, you can simplify it by finding equivalent capacitors for each part, and then reverse the process to find the charge and potential on each individual capacitor.

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Study Notes

Fundamentals of Physics - Chapter 25: Capacitance

  • Capacitance: A capacitor is two isolated conductors with charges +q and -q. Capacitance (C) is defined by the equation q = CV, where V is the potential difference between the plates.

Learning Objectives - Chapter 25-1: Capacitance

  • Schematic Diagrams: Sketch a circuit diagram showing a parallel-plate capacitor, a battery, and an open or closed switch.
  • Electron Behavior: Explain what happens to conduction electrons in a circuit with a battery, an open switch, and an uncharged capacitor when the switch is closed.
  • Capacitance Relationship: Apply the relationship between charge (q), potential difference (V), and capacitance (C) for a capacitor.

Learning Objectives - Chapter 25-2: Calculating Capacitance

  • Gauss's Law: Explain how Gauss's law is used to find the capacitance of a parallel-plate capacitor.
  • Capacitance Calculations: Calculate the capacitance for a parallel-plate capacitor, cylindrical capacitor, spherical capacitor, and an isolated sphere.

Parallel-Plate Capacitor

  • Electric Field: The electric field between the plates is uniform, assuming large plates and small separation.
  • Charge Relationship: The charge q on either plate is related to the electric field E and the area A. The equation is: q = ε₀ EA.
  • Potential Difference: The potential difference V between the plates is related to the electric field and the plate separation d. The equation is V = Ed.
  • Capacitance Formula: The capacitance formula for a parallel plate capacitor is: C = ε₀A/d.

Cylindrical Capacitor

  • Fringing: Neglect fringing at the end of the cylinders for simpler approximations.
  • Charge-Field Relation: The charge q on each plate is related to the electric field E and the cylinder's length L. The equation is: q = ε₀E(2πrL) where r is the radius.
  • Equation: The capacitance equation for a cylindrical capacitor is: C = 2πε₀L / ln(b/a), where a and b are the inner and outer radii, respectively.

Spherical Capacitor

  • Capacitance Equation: The capacitance of a spherical capacitor is given by: C = 4πε₀(ab)/(b − a).
  • Isolated Sphere: For an isolated sphere, the capacitance is: C = 4πε₀R.

Capacitors in Parallel and Series

  • Parallel Capacitors: Capacitors in parallel have the same potential difference (V). The total charge is the sum of the individual charges: q = q₁ + q₂ + q₃ = (C₁ + C₂ + C₃)V.
  • Equivalent Capacitance (Parallel): The total capacitance when capacitors are in parallel is Ceq = ∑Cj (sum of the individual capacitances).
  • Series Capacitors: Capacitors in series share the same charge (q). The total potential difference is the sum of individual potential differences: V = V₁ + V₂ + V₃ = q(1/C₁ + 1/C₂ + 1/C₃).
  • Equivalent Capacitance (Series): The total capacitance when capacitors are in series is 1/Ceq = ∑(1/Cj) (reciprocal sum of the individual capacitances).

Summary Equations

  • q = CV (definition of capacitance)
  • Parallel Plate Capacitor: C = ε₀A/d
  • Cylindrical Capacitor: C = 2πε₀L / ln(b/a)
  • Spherical Capacitor: C = 4πε₀(ab)/(b − a)
  • Isolated Sphere: C = 4πε₀R
  • Parallel: Ceq = ∑Cj
  • Series: 1/Ceq = ∑(1/Cj)

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