Question

A charge \(q\) in a uniform electric field \(\mathbf{E}_{0}\) experiences a constant force \(\mathbf{F}=q \mathbf{E}_{0}\). (a) Show that this force is conservative and verify that the potential energy of the charge at position \(\mathbf{r}\) is \(U(\mathbf{r})=-q \mathbf{E}_{0} \cdot \mathbf{r}\). (b) By doing the necessary derivatives, check that \(\mathbf{F}=-\nabla U\).

Short answer

Force is conservative. Potential: \(U(\mathbf{r}) = -q \mathbf{E}_{0} \cdot \mathbf{r}\), \(\mathbf{F} = -\nabla U\).

Step-by-step solution

Understand the Problem

We need to demonstrate that the force \( \mathbf{F} = q \mathbf{E}_{0} \) is conservative and show that the potential energy is \( U(\mathbf{r}) = -q \mathbf{E}_{0} \cdot \mathbf{r} \). Additionally, we should verify the relationship \( \mathbf{F} = -abla U \).

Define a Conservative Force

A force is considered conservative if the work done by the force around any closed path is zero. Equivalently, a force is conservative if there exists a scalar potential energy function \( U \) such that the force can be written as \( \mathbf{F} = -abla U \).

Propose the Potential Energy Function

Assume the potential energy function \( U(\mathbf{r}) = -q \mathbf{E}_{0} \cdot \mathbf{r} \). This form is suggested because it represents the standard approach to define potential energy in a uniform electric field.

Evaluate the Gradient of U

Calculate the gradient \( abla U \) where \( U(\mathbf{r}) = -q \mathbf{E}_{0} \cdot \mathbf{r} \). The gradient operator is applied to each component of \( \mathbf{r} \), and \( \mathbf{E}_{0} \) being constant results in \( abla U = -q \mathbf{E}_{0} \).

Check the Force from Gradient

From the relationship \( \mathbf{F} = -abla U \), substitute for \( abla U \) giving \( \mathbf{F} = q \mathbf{E}_{0} \). This is consistent with the given \( \mathbf{F} = q \mathbf{E}_{0} \), confirming the force as conservative.

Verify the Conservative Condition

As \( \mathbf{F} = -abla U \) holds true, it implies that \( \mathbf{F} \) is a conservative force. Thus, the work done in a closed path in a uniform electric field is zero, consistent with the properties of conservative forces.

Key concepts

Electric Potential Energy

Electric potential energy is a concept that helps us understand how charged particles interact with electric fields. When a charge is placed in an electric field, it experiences a force which can do work on the charge. This interaction is quantified by the electric potential energy.

The electric potential energy of a charge in a uniform electric field is given by the expression:

• \( U(\mathbf{r}) = -q \mathbf{E}_{0} \cdot \mathbf{r} \)

Here, \( q \) is the charge, \( \mathbf{E}_{0} \) is the uniform electric field, and \( \mathbf{r} \) is the position vector of the charge with respect to a reference point. The negative sign indicates that the electric potential energy decreases when the charge moves in the direction of the field.

This formula allows us to calculate the potential energy for any position of the charge within the field. Understanding this helps to predict how the charge will behave as it moves through the electric field.

Uniform Electric Field

A uniform electric field is one where the electric field strength and direction are constant at every point in the space between two parallel plates or any situation designed to create such a field. This consistency makes it relatively easy to work with mathematically and practically.

In the case of a uniform electric field:

• The field lines are parallel and equally spaced.
• For any charge entering the field, the force experienced remains constant, given by \( \mathbf{F}=q \mathbf{E}_{0} \).
• This force acts in the direction of the electric field if the charge is positive, and opposite if the charge is negative.

Electric fields are often represented by vectors in physics problems, where the vector’s length corresponds to the field's strength. In a uniform field, the implications for forces and movements are simple, because their behavior is quite predictable. A great way to visualize it is as a flat, even surface pushing on all charges equally.

Gradient Operator

The gradient operator, denoted by \( abla \), is a key mathematical tool used in physics, especially when dealing with concepts like electric potential energy. It works on scalar fields, transforming them into vector fields. Physically, this translation helps to determine the direction and rate of increase of a field's strength.

Applied to a potential energy function such as \( U(\mathbf{r}) = -q \mathbf{E}_{0} \cdot \mathbf{r} \), the gradient tells us how the energy changes with position:

• Calculate \( abla U \) to find the force \( \mathbf{F} = - abla U \).
• The result \( abla U = -q \mathbf{E}_{0} \) shows the force's magnitude and direction.
• The negative sign in \( -abla U \) ensures that the force acts in the direction to reduce potential energy, characteristic of conservative forces.

Using the gradient in this way relates the abstract concept of potential energy to the more tangible notion of force. This relationship is vital for understanding how forces in fields cause motion, highlighting the predictive power of physics.