91Ó°ÊÓ

Capacitance refers to the ability of a system to store electrical charge. In this exercise, we have two concentric spherical shells acting as a capacitor with a dielectric medium between them. The capacitance (\( C \)) of such a configuration depends on several factors:

• The radii of the inner (\( a \)) and outer (\( b \)) spheres.
• The dielectric constant (\( \kappa \)) of the material between the spheres.
• Permittivity of free space (\( \varepsilon_0 \)), a constant value approximately equal to \( 8.85 \times 10^{-12} \, \text{F/m} \).

The formula to calculate capacitance is \[C = 4 \pi \varepsilon_0 \kappa \left(\frac{ab}{b-a}\right)\]where \( a \) and \( b \) are the radii converted into meters. By understanding this relationship, you can see how different factors like size and material properties affect the capacitance of a system.

The dielectric constant, symbolized as \( \kappa \), is a measure of a substance's ability to insulate charges from each other. It is a crucial factor in determining the capacitance of a capacitor when a dielectric material separates the conductors. A higher \( \kappa \) implies that the material can store more electrical energy for the same electric field.

In the context of the concentric spheres, the dielectric constant increases the effective capacitance compared to having just air in-between. The greater the dielectric constant, the less electric field required to store a particular amount of charge, showcasing its insulating property. In this exercise, the dielectric constant is given as \( 23.5 \), indicating a strong insulating capability which contributes to a higher capacitance than if the space were filled with air.

Concentric spheres refer to a configuration where one sphere is placed inside another with both sharing the same center. This setup is a common way to design spherical capacitors in electrostatic studies.

• The inner sphere serves as one conductor, and the outer sphere serves as another, holding potential differences between them.
• The space between the spheres can be vacant or filled with a dielectric material.

For this exercise, the radii of the two spheres are given as \( b = 1.70 \, \text{cm} \) and \( a = 1.20 \, \text{cm} \). Understanding this formation helps visualize how electric fields interact within the spherical geometry, and how capacitance is calculated based on the radii and the intervening dielectric.

Induced charge refers to the redistribution of free charge on the surface of a conductor in response to a nearby electric field. This is a fundamental concept in electrostatics, playing a pivotal role when a dielectric is involved.

In this problem, the inner shell of the spherical capacitor experiences an induced charge due to the presence of a dielectric. The induced charge \( q' \) is calculated by the formula:\[q' = q \left( 1 - \frac{1}{\kappa} \right)\]where \( q \) is the free charge on the shell, and \( \kappa \) is the dielectric constant.

• The induced charge impacting the surface influences the overall electrostatic behavior of the system.
• This calculation elucidates how not only free charges contribute to the electric field, but also how the induced charges due to polarization of the dielectric material align themselves.

Comprehending induced charges helps in understanding the efficiency and charge storage capability in capacitors enhanced by dielectric materials.