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Ch 26: Potential and Field
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 26: Potential and FieldProblema 81b
Capitolo 26, Problema 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?

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1
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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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.
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Capacitors & Capacitance (Intro)

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.
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Refraction at Spherical Surfaces

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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Dielectric Breakdown
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