Ch 26: Potential and Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 26: Potential and Field

Problem 81b

Chapter 26, Problem 81b

A spherical capacitor with a 1.0 mm gap between the shells has a capacitance of 100 pF. What are the diameters of the two spheres?

Verified step by step guidance

Understand the formula for the capacitance of a spherical capacitor: $ C = \frac{4 \pi \varepsilon_0}{\frac{1}{r_1} - \frac{1}{r_2}} $, where $ r_1 $ and $ r_2 $ are the radii of the inner and outer spheres, respectively, and $ \varepsilon_0 $ is the permittivity of free space.

Rearrange the formula to solve for $ r_1 $ and $ r_2 $: $ \frac{1}{r_1} - \frac{1}{r_2} = \frac{4 \pi \varepsilon_0}{C} $. Note that the gap between the spheres is given as 1.0 mm, so $ r_2 - r_1 = 1.0 \times 10^{-3} $ m.

Substitute the given values: $ C = 100 \times 10^{-12} $ F and $ \varepsilon_0 = 8.85 \times 10^{-12} $ F/m. This allows you to calculate $ \frac{4 \pi \varepsilon_0}{C} $.

Use the relationship $ r_2 = r_1 + 1.0 \times 10^{-3} $ m to express $ r_2 $ in terms of $ r_1 $. Substitute this into the rearranged formula to create an equation involving only $ r_1 $.

Solve the equation for $ r_1 $, then use $ r_2 = r_1 + 1.0 \times 10^{-3} $ m to find $ r_2 $. Finally, calculate the diameters of the spheres as $ 2r_1 $ and $ 2r_2 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. It is measured in farads (F) and is defined by the formula C = Q/V, where C is capacitance, Q is the charge stored, and V is the voltage across the capacitor. In spherical capacitors, the capacitance depends on the geometry of the spheres and the dielectric medium between them.

Spherical Capacitor

A spherical capacitor consists of two concentric spherical conductors separated by an insulating gap. The capacitance of a spherical capacitor can be calculated using the formula C = 4πε₀(R₁R₂)/(R₂ - R₁), where R₁ and R₂ are the radii of the inner and outer spheres, respectively, and ε₀ is the permittivity of free space. The geometry significantly influences the capacitor's ability to store charge.

Dielectric Gap

The dielectric gap in a capacitor is the space between the two conductive plates or shells, which can affect the capacitance. In the case of a spherical capacitor, the gap is the distance between the inner and outer spheres. A smaller gap generally increases capacitance, as it allows for a stronger electric field and greater charge storage capability, assuming the dielectric material remains constant.

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Related Practice

Textbook Question

Consider a uniformly charged sphere of radius R and total charge Q. The electric field E_out outside the sphere (r≥R) is simply that of a point charge Q. In Chapter 24, we used Gauss’s law to find that the electric field E_in inside the sphere (r≤R) is radially outward with field strength $E i n = \frac{1}{4 π ϵ_{0}} \frac{Q}{R^{3}} r$ . What is the ratio V_center/V_surface?

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Textbook Question

Charge is uniformly distributed with charge density ρ inside a very long cylinder of radius R. Find the potential difference between the surface and the axis of the cylinder.

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Textbook Question

Each capacitor in FIGURE CP26.83 has capacitance C. What is the equivalent capacitance between points a and b?

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Textbook Question

Find an expression for the capacitance of a spherical capacitor, consisting of concentric spherical shells of radii R₁ (inner shell) and R₂ (outer shell).

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