Problem 44 - College Physics

Question

A capacitor has a capacitance of \(50 \mathrm{pF}\), which increases to \(150 \mathrm{pF}\) when a dielectric material is between its plates. What is the dielectric constant of the material?

Step-by-Step Solution

Answer

The dielectric constant is 3.

Step 1: Identify Given Values

We have a capacitor with an initial capacitance of 50 pF (picofarads) which increases to 150 pF when a dielectric material is added. Let's denote the initial capacitance as \(C_0 = 50\) pF and the capacitance with the dielectric as \(C = 150\) pF.

Step 2: Understand the Relationship

The dielectric constant, often symbolized as \(\kappa\) (kappa), is the factor by which the capacitance of a capacitor increases when the dielectric is inserted between its plates. The relationship is given by: \[ C = \kappa \times C_0 \]where \(C\) is the capacitance with the dielectric, and \(C_0\) is the initial capacitance.

Step 3: Rearrange the Formula

We can rearrange the formula to solve for the dielectric constant \(\kappa\):\[ \kappa = \frac{C}{C_0} \]

Step 4: Calculate the Dielectric Constant

Substitute the given values into the formula:\[ \kappa = \frac{150\, \text{pF}}{50\, \text{pF}} \]Simplify this expression to find \(\kappa\).

Step 5: Solve for Kappa

Calculate the fraction:\[ \kappa = 3 \] Thus, the dielectric constant of the material is 3.

Key Concepts

Capacitance

Capacitance is a fundamental concept in the study of electricity and magnetism, particularly when dealing with capacitors. It refers to the ability of a system to store charge per unit voltage. When we talk about capacitance, it is usually measured in farads (F), but one farad is quite large, so smaller units like picofarads (pF) are often used.

• Capacitance indicates how much electric charge a capacitor can hold at a given voltage.
• The basic formula for capacitance is given by: \[ C = \frac{Q}{V} \]where \( C \) is the capacitance, \( Q \) is the charge, and \( V \) is the voltage.

Understanding capacitance helps us in designing circuits that store energy efficiently. In our exercise, the change in capacitance from 50 pF to 150 pF illustrates how a dielectric material affects a capacitor's capacity to hold charge.

Capacitor

A capacitor is an electronic component that stores and releases electricity. It does this by holding charge in an electric field between a pair of conducting plates.

• Standard capacitors typically consist of two plates separated by an insulating material known as a dielectric.
• The ability of a capacitor to store charge depends on the surface area of the plates, the distance between them, and the dielectric material used.

Capacitors are essential in all sorts of electronic devices, where they are used to regulate power supply, filter signals, and maintain voltage levels. When a dielectric material is added between the plates, the capacitor's capacitance is enhanced, as seen in the exercise where the capacitance increases from 50 pF to 150 pF.

Dielectric Material

A dielectric material is an insulating substance that enhances a capacitor's ability to store electric charge. These materials do not conduct electricity themselves but are critical in increasing the capacitance of a capacitor.

• They increase capacitance by polarizing in the presence of an electric field, which reduces the field strength between the plates.
• This action allows the capacitor to hold more charge at the same voltage.
• The dielectric constant, \( \kappa \), is a measure of how much a dielectric material can increase the capacitance of a capacitor compared to air or vacuum.

In your problem, inserting a dielectric material increased the capacitance by a factor of 3, which is the dielectric constant. This constant is an essential characteristic of dielectric materials, helping in the selection and use of materials for specific applications.

Electricity and Magnetism

Electricity and magnetism are interrelated phenomena associated with the presence and motion of electric charge. Capacitors play a crucial role in various electrical circuits and systems, thanks to their ability to store and release energy quickly.

• In circuits, capacitors can serve to smooth fluctuations in power supply, filter out noise, and trigger electrical components.
• The concepts of capacitance and dielectric materials are directly tied to how electric fields are managed within these devices.
• Electric fields created between capacitor plates establish the foundational principles of how capacitors work, influencing how charge is stored and transferred within systems.

Understanding electricity and magnetism, along with how capacitors and dielectrics function, is vital for working with electronic circuits. Such understanding illustrates how simple components can have significant impacts on the efficiency and performance of electrical devices.