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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

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

Step by step solution

01

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 \( \mathbf{F} = -abla U \).
02

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 \( \mathbf{F} = -abla U \).
03

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.
04

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 \( abla U = -q \mathbf{E}_{0} \).
05

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.
06

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In one dimension, it is obvious that a force obeying Hooke's law is conservative (since \(F=-k x\) depends only on the position \(x,\) and this is sufficient to guarantee that \(F\) is conservative in one dimension). Consider instead a spring that obeys Hooke's law and has one end fixed at the origin, but whose other end is free to move in all three dimensions. (The spring could be fastened to a point in the ceiling and be supporting a bouncing mass \(m\) at its other end, for instance.) Write down the force \(\mathbf{F}(\mathbf{r})\) exerted by the spring in terms of its length \(r\) and its equilibrium length \(r_{\mathrm{o}} .\) Prove that this force is conservative. [Hints: Is the force central? Assume that the spring does not bend.]

For a system of \(N\) particles subject to a uniform gravitational field g acting vertically down, prove that the total gravitational potential energy is the same as if all the mass were concentrated at the center of mass of the system; that is, \(U=\sum_{\alpha} U_{\alpha}=M g Y\) where \(M=\sum m_{\alpha}\) is the total mass and \(\mathbf{R}=(X, Y, Z)\) is the position of the \(\mathrm{CM},\) with the \(y\) coordinate measured vertically up. [Hint: We know from Problem 4.5 that \(\left.U_{\alpha}=m_{\alpha} g y_{\alpha} .\right]\)

Calculate the gradient \(\nabla f\) of the following functions, \(f(x, y, z):\) (a) \(f=x^{2}+z^{3} .\) (b) \(f=k y\), where \(k\) is a constant. (c) \(f=r \equiv \sqrt{x^{2}+y^{2}+z^{2}} .\) [Hint: Use the chain rule.] (d) \(f=1 / r\).

Verify that the gravitational force \(-G M m \hat{\mathbf{r}} / r^{2}\) on a point mass \(m\) at \(\mathbf{r},\) due to a fixed point mass \(M\) at the origin, is conservative and calculate the corresponding potential energy.

Near to the point where I am standing on the surface of Planet \(X\), the gravitational force on a mass \(m\) is vertically down but has magnitude \(m \gamma y^{2}\) where \(\gamma\) is a constant and \(y\) is the mass's height above the horizontal ground. (a) Find the work done by gravity on a mass \(m\) moving from \(\mathbf{r}_{1}\) to \(\mathbf{r}_{2}\), and use your answer to show that gravity on Planet \(X,\) although most unusual, is still conservative. Find the corresponding potential energy. (b) Still on the same planet, I thread a bead on a curved, frictionless, rigid wire, which extends from ground level to a height \(h\) above the ground. Show clearly in a picture the forces on the bead when it is somewhere on the wire. (Just name the forces so it's clear what they are; don't worry about their magnitude.) Which of the forces are conservative and which are not? (c) If I release the bead from rest at a height \(h\), how fast will it be going when it reaches the ground?

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