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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?
What happens to the capacitance of a cylindrical capacitor if the radius of the inner cylinder is increased?
For capacitors connected in series, which statement regarding their charge is true?
For capacitors connected in series, which statement regarding their charge is true?
What is the effect on capacitance when the radius of the outer spherical shell of a spherical capacitor is increased?
What is the effect on capacitance when the radius of the outer spherical shell of a spherical capacitor is increased?
What is the characteristic of the potential applied to capacitors that are connected in parallel?
What is the characteristic of the potential applied to capacitors that are connected in parallel?
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When one charged capacitor is connected in parallel with uncharged capacitors, what occurs when equilibrium is reached?
When one charged capacitor is connected in parallel with uncharged capacitors, what occurs when equilibrium is reached?
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What is the relationship between the charge q, capacitance C, and potential difference V in a capacitor?
What is the relationship between the charge q, capacitance C, and potential difference V in a capacitor?
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In a parallel-plate capacitor, what happens to the charges on the plates when the circuit is connected and the switch is closed?
In a parallel-plate capacitor, what happens to the charges on the plates when the circuit is connected and the switch is closed?
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Which of the following best describes the electric field in a uniform parallel-plate capacitor?
Which of the following best describes the electric field in a uniform parallel-plate capacitor?
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What does the presence of a battery in a circuit with a capacitor imply about the potential difference?
What does the presence of a battery in a circuit with a capacitor imply about the potential difference?
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In the schematic diagram of a capacitor circuit, what is the role of the switch?
In the schematic diagram of a capacitor circuit, what is the role of the switch?
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What is the relationship between the electric field E and the charge q for a parallel-plate capacitor according to Gauss' law?
What is the relationship between the electric field E and the charge q for a parallel-plate capacitor according to Gauss' law?
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In the context of a parallel-plate capacitor, what assumption allows us to consider the electric field E as constant between the plates?
In the context of a parallel-plate capacitor, what assumption allows us to consider the electric field E as constant between the plates?
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What expression represents the potential difference V between the plates of a capacitor when integrating the electric field E?
What expression represents the potential difference V between the plates of a capacitor when integrating the electric field E?
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How is capacitance C defined for a parallel-plate capacitor using charge q and potential difference V?
How is capacitance C defined for a parallel-plate capacitor using charge q and potential difference V?
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Which factor is NOT considered when calculating the capacitance of a parallel-plate capacitor?
Which factor is NOT considered when calculating the capacitance of a parallel-plate capacitor?
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What is the formula for the capacitance of a cylindrical capacitor given its length L and inner and outer radii a and b?
What is the formula for the capacitance of a cylindrical capacitor given its length L and inner and outer radii a and b?
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Which of the following expressions represents the electric field E inside a cylindrical capacitor?
Which of the following expressions represents the electric field E inside a cylindrical capacitor?
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For a spherical capacitor, which of the following formulas is used to determine its capacitance?
For a spherical capacitor, which of the following formulas is used to determine its capacitance?
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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?
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?
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What is the capacitance of an isolated sphere with radius R?
What is the capacitance of an isolated sphere with radius R?
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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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Description
Explore the concepts of capacitance in this quiz based on Chapter 25 of Fundamentals of Physics. You'll learn about circuit diagrams of capacitors, electron behavior in circuits, and the application of Gauss's law. Test your knowledge on calculating capacitance and understanding its relationship with charge and voltage.
