Problem 22 - College Physics

Question

An electron is acted on by an electric force of \(3.2 \times 10^{-14} \mathrm{~N}\). What is the magnitude of the electric field at the electron's location?

Step-by-Step Solution

Answer

The magnitude of the electric field is \( 2.0 \times 10^{5} \, \mathrm{N/C} \).

Step 1: Identify the Relationship

The force experienced by a charged particle in an electric field is given by the formula \( F = qE \), where \( F \) is the force, \( q \) is the charge, and \( E \) is the electric field.

Step 2: Know the Charge of an Electron

The charge of an electron is \( q = -1.6 imes 10^{-19} \, \mathrm{C} \).

Step 3: Rearrange the Formula

We need to find the magnitude of the electric field, so rearrange the formula to \( E = \frac{F}{|q|} \). Here, we take the absolute value of \( q \) because we are only finding the magnitude, not the direction.

Step 4: Substitute the Values

Substitute the given force and the charge into the equation: \[ E = \frac{3.2 imes 10^{-14} \, \mathrm{N}}{1.6 imes 10^{-19} \, \mathrm{C}} \].

Step 5: Calculate the Electric Field

Performing the division gives \[ E = 2.0 imes 10^{5} \, \mathrm{N/C} \]. This is the magnitude of the electric field.

Key Concepts

Force on Charged Particle

When a charged particle, such as an electron, is placed in an electric field, it experiences a force. This force can cause the particle to move, and its magnitude depends on two main factors: the size of the charge on the particle and the strength of the electric field it is in. The direction of the force is determined by the nature of the charge:

• A positive charge experiences a force in the direction of the electric field.
• A negative charge, like an electron, experiences a force opposite to the direction of the electric field.

This concept is fundamental in understanding how charged particles react to electric fields, which is essential in fields like electronics and physics.

Electric Force Formula

The electric force acting on a charged particle is calculated using the formula \( F = qE \):

• \( F \) is the force measured in newtons \( \mathrm{N} \).
• \( q \) is the charge in coulombs \( \mathrm{C} \).
• \( E \) is the electric field in newtons per coulomb \( \mathrm{N/C} \).

This equation tells us that the force is directly proportional to both the charge and the electric field. If either the charge or the electric field increases, so does the force proportionally. This principle is widely used in designing and understanding various electrical devices, from simple circuits to complex machinery.

Electron Charge

The charge of an electron is a fundamental constant in physics. It is denoted as \( q = -1.6 \times 10^{-19} \, \mathrm{C} \). This negative sign indicates that electrons are negatively charged particles. In calculations involving magnitude, we often use the absolute value, ignoring the negative sign, to focus solely on how much charge there is.

• This constant value helps in calculating forces in any scenario involving electrons.
• In practical applications, this charge is crucial for determining behaviors in electrical and electronic systems, such as in semiconductor devices.

Electric Field Magnitude

The magnitude of an electric field is a measurement of how strong the field is at a particular point in space. It is calculated once we rearrange the formula for electric force, focusing on the electric field:\[ E = \frac{F}{|q|} \]Where:

• \( E \) is the electric field magnitude in \( \mathrm{N/C} \).
• \( F \) is the force in newtons \( \mathrm{N} \).
• \(|q|\) is the absolute value of the charge \( \mathrm{C} \).

In practical terms, this formula helps you figure out how strong the electric field is around a charged object, simplifying the problem to a matter of division. Understanding the strength of the field is key to predicting how charged particles will behave, an application used in various technological and research fields.