Problem 11 - College Physics

Question

Two large metal parallel plates carry opposite charges of equal magnitude. They are separated by \(45.0 \mathrm{mm},\) and the potential difference between them is 360 \(\mathrm{V}\) (a) What is the magnitude of the electric field (assumed to be uniform) in the region between the plates? (b) What is the magnitude of the force this field exerts on a particle with charge \(+2.40 \mathrm{nC}\) ?

Step-by-Step Solution

Answer

(a) 8000 V/m, (b) 1.92 x 10^-5 N

Step 1: Understand the Problem

We have a pair of parallel plates with a potential difference of 360 V and separated by a distance of 45.0 mm. We need to calculate the magnitude of the electric field between the plates and the force on a particle with charge +2.40 nC in this field.

Step 2: Calculate the Electric Field

The electric field (E) between two parallel plates is given by the formula: \[ E = \frac{V}{d} \]where V is the potential difference and d is the separation between the plates. Convert the separation distance from mm to meters: \[ d = 45.0 \text{ mm} = 0.045 \text{ m} \]Now, substitute the values into the formula: \[ E = \frac{360 \text{ V}}{0.045 \text{ m}} \approx 8000 \text{ V/m} \]

Step 3: Calculate the Force on the Particle

The force (F) exerted on a charged particle in an electric field is given by: \[ F = qE \]where q is the charge of the particle and E is the electric field. Substitute the given values:\[ q = 2.40 \text{ nC} = 2.40 \times 10^{-9} \text{ C} \] (convert nC to C)Substitute into the equation:\[ F = (2.40 \times 10^{-9} \text{ C})(8000 \text{ V/m}) = 1.92 \times 10^{-5} \text{ N} \]

Step 4: Conclude the Results

The magnitude of the electric field between the plates is 8000 V/m. The force exerted by this electric field on a particle with charge +2.40 nC is approximately 1.92 x 10^-5 N.

Key Concepts

Parallel Plates

Parallel plates are an essential concept in physics, particularly in the study of electric fields. Imagine two large, flat, metal plates that are placed closely together and carry opposite charges. They are called "parallel" because they are aligned in such a way that their surfaces are parallel to each other.

This configuration creates a uniform electric field in the region between the plates, making calculations straightforward. The uniformity means the field lines are straight, equally spaced, and perpendicular to the plates. This is advantageous when it comes to calculating electrical properties, as the uniform field can be described with simple mathematics.

The study of parallel plates helps us understand concepts such as capacitors, which are devices designed to store electrical energy. These plates create a consistent field that can hold charge and release it when necessary. This fundamental property is what allows many everyday electronic devices to function.

Potential Difference

Potential difference, often referred to as voltage, is a key term when discussing electric fields between parallel plates. It signifies the work needed to move a charge from one plate to another.

In simpler terms, potential difference is the amount of energy per unit charge required to move a charge within an electric field. In our exercise, the potential difference between the plates is given as 360 volts (V). This tells us how strong the field is in pulling or pushing a charge across the space between the plates.

Potential difference is measured in volts and is calculated using the formula:

• V = E d

where V represents the potential difference, E is the magnitude of the electric field, and d is the distance between the plates. In practical applications, understanding voltage is crucial because it determines how much energy is required to move charges in a circuit, affecting everything from how bright a light bulb glows to how fast a motor spins.

Charge

Charge is a fundamental property of matter that causes it to experience a force when placed in an electric field. It can be either positive or negative, corresponding to two types of electric charge: protons carry positive charge, while electrons carry negative charge.

In the problem we explored, a small particle with a positive charge of +2.40 nanocoulombs (nC) is situated between the parallel plates. This charge will interact with the electric field created by the plates.

Charges are interesting because they do not occur in isolation; similar charges repel each other, while opposite charges attract. The behavior of charges underlies many phenomena in nature and technology. When dealing with electrical circuits and fields, the movement and interaction of these charges are what enables electronic devices to operate.

Understanding charge helps us predict how particles will move and interact within fields, which is crucial for designing circuits and understanding electrical interactions.

Electric Force

Electric force is the force exerted by an electric field on a charged particle. This force can either attract or repel the particle, depending on the type of charge it possesses.

In our context, the problem involves determining the force that the uniform electric field between two parallel plates exerts on a charged particle. This force is calculated using the formula:

• F = q E

where F is the force, q is the charge of the particle, and E is the electric field.

For instance, in the given problem, once we determine the electric field to be 8000 V/m and the charge to be +2.40 nC, we can find the force as approximately 1.92 x 10^-5 newtons (N). This force is what would push or pull the charged particle within the field.

Understanding the electric force is crucial for predicting and manipulating the behavior of charged particles in various applications, from designing electronic components to developing electric propulsion systems.