Problem 25 - University Physics with Modern Physics

Question

A positive charge \(q\) is fixed at the point \(x=0, y=0,\) and a negative charge \(-2 q\) is fixed at the point \(x=a, y=0\) . (a) Show the positions of the charges in a diagram. (b) Derive an expression for the potential \(V\) at points on the \(x\) -axis as a function of the coordinate \(x\) . Take \(V\) to be zero at an infinite distance from the charges. (c) At which positions on the \(x\) -axis is \(V=0 ?(\text { d) Graph }\) \(V\) at points on the \(x\) -axis as a function of \(x\) in the range from \(x=-2 a\) to \(x=+2 a\) (e) What does the answer to part \((b)\) become when \(x \gg a ?\) Explain why this result is obtained.

Step-by-Step Solution

Answer

The potential is zero at \( x = -a \) and \( x = \frac{3a}{2} \) on the x-axis.

Step 1: Diagram the charges

To show the positions of the charges in a diagram, plot the positive charge \( q \) at the origin \((x=0, y=0)\) and the negative charge \(-2q\) at the point \((x=a, y=0)\) on a coordinate plane. The two charges lie on the x-axis.

Step 2: Derive the potential expression

The electric potential at a point \( x \) on the x-axis due to a charge \( q \) at position \( x_0 \) is given by \( V = \frac{k q}{|x - x_0|} \). Therefore, for the charge \( q \) at \( x=0 \), the potential is \( V_1 = \frac{k q}{|x|} \), and for the charge \(-2q\) at \( x=a \), the potential is \( V_2 = \frac{-2k q}{|x-a|} \). The total potential \( V \) at any point \( x \) is then \( V = V_1 + V_2 = \frac{k q}{|x|} + \frac{-2k q}{|x-a|} \).

Step 3: Find where potential V = 0

Set the total potential expression to zero: \( \frac{k q}{|x|} + \frac{-2k q}{|x-a|} = 0 \). Simplify to find: \( \frac{1}{|x|} = \frac{2}{|x-a|} \). Solving gives two cases: \( x = -a \) and \( x = \frac{3a}{2} \). These are the positions on the x-axis where the potential is zero.

Step 4: Graph V as a function of x

To plot the graph of potential \( V \) as a function of \( x \) from \( x=-2a \) to \( x=2a \), calculate \( V \) at various points in this range using the expression derived in Step 2. The graph will show the potential approaching zero far from the charges and having changes in sign due to the different contributions of each charge as they are approached.

Step 5: Analyze V for x much greater than a

When \( x \gg a \), both \( x \) and \( x-a \) are approximately equal to \( x \). The potential simplifies to \( V \approx \frac{k q}{x} - \frac{2k q}{x} = \frac{-k q}{x} \). This occurs because the negative charge dominates; far from the origin, it behaves like a single charge. Thus, \( V \) approximates the potential of a single effective charge \(-q\) positioned far along \( x \).

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes the force between two point charges. The law states that the electric force \( F \) between two charges is directly proportional to the product of the charges' magnitudes and inversely proportional to the square of the distance between their centers. The mathematical expression for Coulomb's Law is given by:

• \( F = k \frac{|q_1 q_2|}{r^2} \) where \( F \) is the force, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant.

The direction of the force is along the line joining the charges:

• If both charges have the same sign, the force is repulsive, pushing them apart.
• If the charges have opposite signs, the force is attractive, pulling them together.

Understanding this concept helps us predict how charges interact at a distance and is foundational for many topics in electromagnetism.

Point Charges

Point charges are idealized models of charges used in physics to simplify problems involving electric fields and forces. They are considered to have no physical dimensions; instead, all of the charge is assumed to be concentrated at a single point in space. This abstraction is useful because:

• It allows for easy application of Coulomb's Law, as seen in our original problem where the charges were fixed at certain locations along the x-axis.
• Point charges create radial electric fields, which are symmetrical and point directly towards or away from the charge, depending on its sign.

Despite being theoretical, point charges offer a great way to approach complex scenarios, such as calculating the electric field or electric potential of distributed charges by considering them as a series of point charges.

Electric Field

The electric field is a concept that describes the influence a charge exerts on other charges in its vicinity. It is a vector field, meaning it has both magnitude and direction at every point in space. The strength of the electric field \( E \) produced by a point charge \( q \) at a distance \( r \) from the charge is given by:

• \( E = k \frac{|q|}{r^2} \)

• The direction of the electric field is radially outward for positive charges and radially inward for negative charges.

The electric field represents the force per unit charge that a positive test charge would experience if it were placed in the field. This concept is especially useful in determining how electric charges influence each other without directly considering their interactions, as electric field lines can be sketched to visualize the field's behavior, especially when dealing with multiple charges, like in our exercise.Understanding electric fields helps in visualizing the complex interactions in electrostatics and simplifying the calculation of forces and potentials.