Problem 32 - College Physics

Question

\(\bullet\) (a) What must the charge (sign and magnitude) of a 1.45 \(\mathrm{g}\) particle be for it to remain balanced against gravity when placed in a downward-directed electric field of magnitude 650 \(\mathrm{N} / \mathrm{C}^{?}\) (b) What is the magnitude of an electric field in which the electric force it exerts on a proton is equal in magni- tude to the proton's weight?

Step-by-Step Solution

Answer

(a) Charge is \(-2.186 \times 10^{-5} \ C\); (b) Electric field is \(1.025 \times 10^{-7} \ N/C\).

Step 1: Understand the Forces

For the particle to remain balanced against gravity, the electric force acting on it must be equal and opposite to the gravitational force. The gravitational force can be calculated using the formula \( F_g = mg \), where \( m \) is the mass and \( g \) is the acceleration due to gravity \( (9.8 \ m/s^2) \).

Step 2: Calculate the Gravitational Force

Given the mass of the particle \( m = 1.45 \ g \) \( (0.00145 \ kg) \), the gravitational force \( F_g = 0.00145 \times 9.8 = 0.01421 \ N \).

Step 3: Apply Electric Force Equation

The electric force \( F_e \) can be calculated with \( F_e = qE \) where \( q \) is the charge on the particle and \( E = 650 \ N/C \) is the electric field. Since \( F_e = F_g \), we can set \( qE = mg \).

Step 4: Solve for Charge

Rearrange the equation to solve for \( q \):\[ q = \frac{mg}{E} = \frac{0.01421}{650} = 2.186 \times 10^{-5} \ C \]. The charge must be negative because the electric field is downward-directed and should balance the upward gravitational force.

Step 5: Calculate Proton Weight

The weight of a proton is given by \( F_g = mg \), with the mass of the proton \( m = 1.67 \times 10^{-27} \ kg \). Thus, \( F_g = 1.67 \times 10^{-27} \times 9.8 = 1.64 \times 10^{-26} \ N \).

Step 6: Determine Electric Field for Proton

The electric force on the proton should equal its weight, so \( q_pE = 1.64 \times 10^{-26} \ N \), where \( q_p = 1.6 \times 10^{-19} \ C \). Solving for \( E \):\[ E = \frac{1.64 \times 10^{-26}}{1.6 \times 10^{-19}} = 1.025 \times 10^{-7} \ N/C \].

Key Concepts

Gravitational Force

Gravitational force is a fundamental concept in physics. It is the force exerted by the Earth's gravity on any object with mass. To calculate this force, we use the formula:

• \( F_g = mg \)

In this equation, \( F_g \) represents the gravitational force, \( m \) stands for the mass of the object, and \( g \) denotes the acceleration due to gravity, which is approximately \( 9.8 \, \text{m/s}^2 \) on the surface of the Earth. Whenever you have an object with mass, gravity pulls it towards the Earth's center with a force that depends directly on its mass. The more massive the object, the stronger the pull.
Understanding gravitational force is essential in balancing other forces, like the electric force, as seen in many physics problems. This balance is crucial in cases where an object needs to stay suspended or neutralized between forces acting on it.

Charge Calculation

Charge calculation involves determining the amount of electric charge needed to balance forces acting on an object in an electric field. When a charged particle is subjected to an electric field, it experiences an electric force. To find the charge \( q \) necessary for balancing a gravitational force, use the formula:

• \( F_e = qE \)
• \( q = \frac{mg}{E} \)

Here, \( F_e \) represents the electric force equal to the gravitational force \( F_g \). \( q \) stands for the charge, and \( E \) is the electric field strength. The calculation becomes straightforward: you solve for \( q \) by dividing the gravitational force by the electric field's magnitude. In the original exercise, they calculated the charge needed to counterbalance gravity in a 650 N/C electric field, resulting in a specific charge value.
Remember, in the scenario where the electric field has a direction, the necessary charge needs to have the opposite sign to ensure balance. In downward electric fields, like those mentioned, a negative charge ensures an upward electric force.

Electric Field

The electric field is an invisible field around charged particles that exerts a force on other charges within the field. It is essential in determining how much force a charge will experience. The formula to calculate the force experienced by a charge in an electric field is:

• \( F_e = qE \)

Where \( F_e \) is the electric force, \( q \) the charge, and \( E \) the electric field strength. Understanding electric fields is critical in physics since these fields naturally arise wherever charges exist.
Moreover, in the original exercise, they calculated the electric field strength required to make the electric force on a proton equal its gravitational force. By using the known charge of a proton \( (1.6 \times 10^{-19} \, C) \) and its gravitational force, the electric field strength was determined easily.
The balance of electric force with gravitational force highlights how electric fields can be manipulated to counterbalance different forces acting on charged particles.