Chapter 24: Problem 13
A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. The inner sphere has radius 15.0 cm and the capacitance is 116 pF. (a) What is the radius of the outer sphere? (b) If the potential difference between the two spheres is 220 V, what is the magnitude of charge on each sphere?
Short Answer
Step by step solution
Understanding Capacitance of a Spherical Capacitor
Rearranging the Formula for the Outer Radius
Inserting Values and Solving for Outer Radius
Calculating the Permittivity Substitution
Solving the Capacitance Equation
Determine the Charge on Each Sphere
Solving the Charge Equation
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Capacitance
In the case of a spherical capacitor, which consists of two concentric spherical conducting shells, the capacitance formula is given by:
\[C = \frac{4\pi\varepsilon_0 \cdot a \cdot b}{b-a}\]
where \( a \) is the radius of the inner sphere, \( b \) is the radius of the outer sphere, and \( \varepsilon_0 \) is the permittivity of free space. In this formula, the difference \( b-a \) represents the separation between the spheres, controlling how electric fields interact. A high capacitance means more charge can be stored at a given potential difference.
Capacitance is also influenced by the material between the plates (or spheres in this case). In our exercise, the space is a vacuum which standardizes calculations using \( \varepsilon_0 \).
Potential Difference
For capacitors, the potential difference is related to the amount of electric charge stored. In mathematical terms, it is described by the formula:
\[V = \frac{Q}{C}\]
where \( Q \) is the electric charge stored in the capacitor and \( C \) is the capacitance. This relation signifies that for a given amount of stored charge, a larger capacitance will lead to a smaller potential difference and vice versa. In our exercise, the potential difference between the spherical shells is significant for calculating the stored charge.
Understanding potential differences is crucial when determining the operation conditions of capacitors in electrical circuits and systems.
Electric Charge
The relationship between charge, capacitance, and potential difference is expressed as:
\[Q = C \cdot V\]
This formula shows that the charge \( Q \) stored in a capacitor is a product of the capacitance \( C \) and the potential difference \( V \) across the capacitor.
In our scenario, to find the charge stored in the spherical capacitor, one would multiply its capacitance (116 pF in the problem) by the potential difference (220 V). The calculation tells us how much charge each sphere of the capacitor holds under the given conditions, highlighting the utility of capacitors in managing electrical energy.
Permittivity of Free Space
This constant plays a central role in determining the capacitance of capacitors, as seen in the capacitance formula for spherical capacitors:
\[C = \frac{4\pi\varepsilon_0 \cdot a \cdot b}{b-a}\]
It essentially relates the geometric factors (radii of the spheres) to the potential to store charge through an electric field in a vacuum.
Understanding \( \varepsilon_0 \) is vital not only in theoretical physics but also in practical applications involving insulators and capacitors. It helps in estimating the efficiency of capacitors in storing energy and ensures precision in computations involving electric and magnetic fields.