Problem 37 - Physics

Question

Interactive Solution \(18.37\) at provides a model for problems of this kind. A small object has a mass of \(3.0 \times 10^{-3} \mathrm{~kg}\) and a charge of \(-34 \mu \mathrm{C}\). It is placed at a certain spot where there is an electric field. When released, the object experiences an acceleration of \(2.5 \times 10^{3} \mathrm{~m} / \mathrm{s}^{2}\) in the direction of the \(+x\) axis. Determine the magnitude and direction of the electric field.

Step-by-Step Solution

Answer

Magnitude: \(2.21 \times 10^{5} \text{ N/C}\), Direction: \(-x\) axis.

Step 1: Understand the Relationship Between Electric Force and Electric Field

The force exerted on a charged object in an electric field is given by the equation \( F = qE \), where \( F \) is the force, \( q \) is the charge, and \( E \) is the electric field. We can rearrange this formula to find the electric field: \( E = \frac{F}{q} \).

Step 2: Calculate the Force Using Newton's Second Law

According to Newton's second law, the force experienced by the object is also given by \( F = ma \), where \( m \) is the mass and \( a \) is the acceleration. Substitute the given values: \( F = (3.0 \times 10^{-3} \text{ kg}) (2.5 \times 10^{3} \text{ m/s}^2) \). This calculates to \( F = 7.5 \text{ N} \).

Step 3: Calculate the Electric Field Magnitude

Substitute the force and charge values into the electric field equation: \( E = \frac{7.5 \text{ N}}{-34 \times 10^{-6} \text{ C}} \). Solving this gives \( E = -2.21 \times 10^{5} \text{ N/C} \).

Step 4: Determine the Direction of the Electric Field

The negative sign in the electric field calculation indicates that the direction of the electric field is opposite to the direction of the force experienced by the negative charge. Since the object accelerates in the \(+x\) direction, the electric field is actually in the \(-x\) direction.

Key Concepts

Electric Force

Electric force acts when a charged object enters an electric field. It is a fundamental interaction where an electric field exerts a force on an object with charge. The magnitude of the force (\( F \)) on the charge (\( q \)) is calculated using the formula:

• \( F = qE \)

Where \( E \) is the electric field. This relationship shows us that the force is directly proportional to the charge.
For example, the more charged the object, the more force it feels in the electric field.
In practice, this means that an object with a larger charge will experience a stronger "push" or "pull" in an electric field. Understanding this basic relationship is crucial to solving problems involving forces and electric fields.

Newton's Second Law

Newton's Second Law describes how an object changes motion under the influence of a force. It is summarized by the formula:

• \( F = ma \)

Where \( F \) is the force applied, \( m \) is the mass of the object, and \( a \) is its acceleration. This equation indicates that force is the product of mass and acceleration.
More simply, heavier objects require more force to achieve the same acceleration as lighter ones.
In problems involving electric forces, we often use this law to connect mechanical and electric scenarios. It allows us to compute unknown quantities like force when mass and acceleration are known.

Charge and Mass Relationship

The relationship between charge and mass can influence how an object behaves in an electric field. While mass determines how an object moves in response to an electric force (via Newton's Second Law), charge determines the magnitude of the force itself. This means:

• Objects with greater charge experience greater force in the same electric field.
• An object's mass impacts how much it accelerates for a given force.

Hence, two charged objects with different masses may experience different accelerations in the same field.
This relationship helps explain why lighter charged objects often move more quickly in electric fields than heavier charged ones.
However, acceleration also depends on the ratio of charge to mass, known as the charge-to-mass ratio, crucial for understanding motion in electric contexts.

Acceleration in Electric Fields

Acceleration in electric fields distinguishes how objects respond to electric forces. When charged particles enter an electric field, they accelerate due to the force exerted on them. The mechanics of this acceleration are explained by Newton's Second Law and the force relation to the electric field:

• The acceleration (\( a \)) is computed as:
• \( a = \frac{F}{m} = \frac{qE}{m} \)

This formula demonstrates that acceleration depends on both the electric field strength and the charge-to-mass ratio (\( \frac{q}{m} \)).
Consequently, a strong electric field or a high charge-to-mass ratio will result in higher acceleration.
Recognizing how these factors interact helps us predict how charged objects will speed up or slow down when placed in an electric field. Electric fields influence motion in a way unique from gravitational fields, highlighting the dynamic nature of charges in physics. This insight assists in understanding various physical and engineering applications, such as the operation of devices that manipulate charged particles.