Ch 24: Capacitance and Dielectrics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Non è quello che usi tu?Cambia libro di testo

Young & Freedman Calc 15th Edition

Ch 24: Capacitance and Dielectrics

Problema 11

Capitolo 24, Problema 11

A spherical capacitor contains a charge of $3.30$ nC when connected to a potential difference of $220$ V. If its plates are separated by vacuum and the inner radius of the outer shell is $4.00$ cm, calculate: (a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.

Guida verificata passo dopo passo

To find the capacitance of a spherical capacitor, use the formula for capacitance:

C = \frac{Q}{V}

, where

Q

is the charge and

V

is the potential difference. Substitute the given values:

= 3.30 imes 10^{-9} \(\text{ C}\)

and

= 220 \(\text{ V}\)

For part (b), use the formula for the capacitance of a spherical capacitor:

= 4\(\text{π}\)\(\text{ε}\)_0 rac{r_1 r_2}{r_2 - r_1}

, where

r_1

is the radius of the inner sphere and

r_2

is the radius of the outer shell. Rearrange the formula to solve for

r_1

Substitute the known values into the rearranged formula:

C

from part (a),

= 0.04 \(\text{ m}\)

, and

= 8.85 imes 10^{-12} \(\text{ F/m}\)

For part (c), calculate the electric field just outside the surface of the inner sphere using the formula:

= rac{Q}{4\(\text{π}\)\(\text{ε}\)_0 r_1^2}

. Substitute the values for

Q

ε_0

, and

r_1

from part (b).

Ensure all units are consistent, converting cm to m where necessary, and verify each calculation step to ensure accuracy in determining the capacitance, radius, and electric field.

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Capacitance of a Spherical Capacitor

Capacitance is a measure of a capacitor's ability to store charge per unit voltage. For a spherical capacitor, it depends on the radii of the inner and outer spheres and the permittivity of the medium between them. The formula for capacitance in a vacuum is C = 4πε₀(r₁r₂)/(r₂ - r₁), where r₁ and r₂ are the radii of the inner and outer spheres, respectively.

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Percorso guidato

03:11

Capacitance of Spherical Capacitor

Electric Field in a Spherical Capacitor

The electric field in a spherical capacitor is determined by the charge on the inner sphere and the distance from the center. Just outside the surface of the inner sphere, the electric field E can be calculated using E = Q/(4πε₀r₁²), where Q is the charge and r₁ is the radius of the inner sphere. This field is radially outward and decreases with distance.

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Percorso guidato

06:13

Intro to Capacitors

Potential Difference and Charge Relationship

The relationship between potential difference (voltage) and charge in a capacitor is given by Q = CV, where Q is the charge, C is the capacitance, and V is the voltage. This equation helps in determining the capacitance when the charge and voltage are known, and is fundamental in analyzing capacitors in circuits.

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Percorso guidato

05:45

Potential Difference Between Two Charges

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