Problem 27 - University Physics with Modern Physics

Question

(a) What must the charge (sign and magnitude) of a 1.45-g particle be for it to remain stationary when placed in a downward-directed electric field of magnitude 650 N\(/\)C? (b) What is the magnitude of an electric field in which the electric force on a proton is equal in magnitude to its weight?

Step-by-Step Solution

Answer

(a) The charge is +2.19 x 10^{-5} C. (b) The electric field is 1.02 x 10^{-7} N/C.

Step 1: Understanding the Forces on the Particle

To keep the particle stationary in an electric field, the upward electric force must balance the weight of the particle. Therefore, the electric force and gravitational force are equal in magnitude but opposite in direction.

Step 2: Calculating the Gravitational Force on the Particle

The gravitational force (\( F_g \)) is given by\[ F_g = m \cdot g \]where\( m = 1.45 \text{ g} = 0.00145 \text{ kg} \) (since 1 g = 0.001 kg) and\( g = 9.8 \text{ m/s}^2 \).Thus,\[ F_g = 0.00145 \text{ kg} \cdot 9.8 \text{ m/s}^2 = 0.01421 \text{ N} \].

Step 3: Calculating the Required Electric Force

The electric force (\( F_e \)) needed to keep the particle stationary is\( F_e = F_g = 0.01421 \text{ N} \).The electric force is also given by\[ F_e = q \cdot E \]where\( E = 650 \text{ N/C} \).Solving for\( q \), we get\( q = \frac{F_e}{E} \).

Step 4: Solving for the Charge of the Particle

Substituting the known values into the equation,\[ q = \frac{0.01421 \text{ N}}{650 \text{ N/C}} = 2.1877 \times 10^{-5} \text{ C} \].The particle must have a positive charge since the electric field is directed downward and the electric force needs to balance gravity which acts downward.

Step 5: Understanding the Force on a Proton

The force on a proton in an electric field is given by\[ F_e = q \cdot E \]where the charge of a proton is\( q = 1.602 \times 10^{-19} \text{ C} \). The gravitational force (weight) on a proton is\( F_g = m \cdot g \), with\( m = 1.67 \times 10^{-27} \text{ kg} \).

Step 6: Calculating the Electric Field for a Proton

We need an electric field such that the electric force\( F_e \)equals the gravitational force\( F_g \).Solving\( q \cdot E = m \cdot g \) for\( E \), we get\( E = \frac{m \cdot g}{q} \).Substitute the values:\( E = \frac{1.67 \times 10^{-27} \text{ kg} \cdot 9.8 \text{ m/s}^2}{1.602 \times 10^{-19} \text{ C}} = 1.02 \times 10^{-7} \text{ N/C} \).

Key Concepts

Electric force

Electric force is a fundamental concept in electrostatics, representing the force between charged objects. When an object carries an electric charge, it can exert force on other charged objects without direct physical contact. This is known as the electric force. It plays a crucial role in scenarios where charged particles or objects interact in electric fields.

The strength of the electric force depends on two main factors:

• The magnitude of the electric charges involved: The greater the charge, the stronger the force.
• The distance between the charges: The closer the objects, the stronger the force.

Electric force is calculated using the formula \( F_e = q \cdot E \), where \( F_e \) is the electric force, \( q \) is the charge, and \( E \) is the electric field strength. This formula indicates that the electric force magnitude directly relates to both the charge and the electric field it is in. In the solved problem, the electric force on the particle needed to counteract gravity and keep it stationary was found using this relationship.

Gravitational force

Gravitational force is the attraction between objects with mass. It is one of the four fundamental forces of nature and is described by Newton's law of universal gravitation. This force is always attractive and pulls masses toward each other. The force's magnitude depends on:

• The masses of the objects involved: More massive objects exert a more significant gravitational force.
• The distance between the objects: The closer the masses, the stronger the gravitational pull.

In basic terms, the gravitational force acting on a small object near Earth's surface is its weight, calculated with \( F_g = m \cdot g \), where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

In the exercise, the gravitational force was used to determine the electric force necessary to keep a charged particle stationary within an electric field. By balancing these forces, we ensure that the particle remains in a state of equilibrium.

Electric charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. It comes in two types: positive and negative. Particles with like charges repel each other, while opposite charges attract.

Charge is usually measured in coulombs (C). Fundamental particles such as electrons and protons carry charge, with electrons holding a negative charge \(-1.602 \times 10^{-19} \, \text{C}\) and protons a positive charge of the same magnitude. Charge conservation is a principle in physics stating that the total electric charge in an isolated system remains constant over time.

In the problem, the charge calculated for the particle needed to counterbalance the gravitational force was found to be positive. This ensures that the upward electric force counteracts the downward pull of gravity, keeping the particle stationary within the downward electric field.

Electric field strength

Electric field strength, or electric field intensity, is a measure of the force per unit charge experienced by a positive test charge placed in the field. This vector quantity indicates both the field's direction and its magnitude. Electric fields are created by electric charges and influence other charges within the field.

The formula for electric field strength is \( E = \frac{F_e}{q} \), where \( E \) is the electric field strength, \( F_e \) is the electric force, and \( q \) is the charge experiencing the force. The unit of electric field strength is newtons per coulomb (N/C).

In the exercise, the electric field needed to balance the force due to a proton's weight was calculated using this formula. A unique electric field was required so the electric force would equal the gravitational force, effectively making the proton's weight and the electric force identical, allowing it to remain at rest in the field.