Chapter 24: Problem 32
A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius 12.5 cm, and the outer sphere has radius 14.8 cm. A potential difference of 120 V is applied to the capacitor. (a) What is the energy density at \(r\) = 12.6 cm, just outside the inner sphere? (b) What is the energy density at \(r\) = 14.7 cm, just inside the outer sphere? (c) For a parallel-plate capacitor the energy density is uniform in the region between the plates, except near the edges of the plates. Is this also true for a spherical capacitor?
Short Answer
Step by step solution
Understand Energy Density Formula
Electric Field in a Spherical Capacitor
Solve for Charge Q
Find Electric Field at r = 12.6 cm
Calculate Energy Density at r = 12.6 cm
Find Electric Field at r = 14.7 cm
Calculate Energy Density at r = 14.7 cm
Compare Energy Densities
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Energy Density
In spherical capacitors, unlike parallel-plate capacitors, the energy density is not uniform because the electric field varies with the distance from the center. This variation is due to the spherical geometry, which affects how the electric field disperses over space.
Electric Field
Here's a quick breakdown of what happens:
- The electric field decreases as you move away from the inner sphere because it is inversely proportional to \( r^2 \).
- This decrease means the electric field is stronger closer to the inner sphere, which affects the energy density calculation.
Potential Difference
- The potential difference is a crucial factor because it determines the electric field's magnitude and direction.
- In practical scenarios, a higher potential difference usually means the capacitor can store more energy, assuming the physical dimensions are constant.
Spherical Shells
- The inner shell (with radius 12.5 cm)
- The outer shell (with radius 14.8 cm)
A key point to remember about spherical capacitors:
- They differ from parallel-plate capacitors mainly due to the shape and arrangement of the electrodes.
- This geometry affects not only the electric field distribution but also the energy density and capacitance.
Permittivity of Free Space
This constant is essential in electrostatics because:
- It determines the ability of the vacuum (or any other material) to resist the formation of an electric field.
- It appears in fundamental equations used to calculate electric fields, forces, and energy densities.