Question

Two positive point charges \(q\) are placed on the \(x\)-axis, one at \(x = a\) and one at \(x = -a\). (a) Find the magnitude and direction of the electric field at \(x = 0\). (b) Derive an expression for the electric field at points on the x-axis. Use your result to graph the \(x\)-component of the electric field as a function of \(x\), for values of \(x\) between \(-\)4a and \(+\)4a.

Short answer

(a) The electric field at \(x = 0\) is zero. (b) \(E_x(x) = \frac{kq}{(x-a)^2} - \frac{kq}{(x+a)^2}\); plot is symmetric spikes at \(x = \pm a\).

Step-by-step solution

Understand the Problem

Two identical point charges, each with charge magnitude \(q\), are placed symmetrically on the \(x\)-axis at \(x = -a\) and \(x = a\). We need to determine the electric field at \(x = 0\) (the origin) and derive an expression for the electric field at any point on the \(x\)-axis. Finally, we plot the \(x\)-component of the electric field for various values of \(x\).

Calculate Electric Field at x = 0

The electric field due to a point charge at a point \(P\) is given by \(E = \frac{kq}{r^2}\), where \(k\) is Coulomb's constant, \(q\) is the charge, and \(r\) is the distance from the charge to the point \(P\). At \(x = 0\), each charge creates an electric field at the origin with magnitude \(E_1 = E_2 = \frac{kq}{a^2}\), both directed along the \(x\)-axis.

Direction of Electric Fields at x = 0

At \(x = 0\), the electric field due to the charge at \(x = a\) points towards the positive \(x\)-direction, while the electric field due to the charge at \(x = -a\) points towards the negative \(x\)-direction. Since both fields have the same magnitude, they cancel each other out.

Conclusion for E at x = 0

At \(x = 0\), the net electric field is \(E = 0\) because the contributions from the two charges cancel each other in both magnitude and direction.

Derive General Expression for Electric Field on x-axis

For a point \(P\) on axis at \(x\), the distances from the charges at \(-a\) and \(a\) are \(|x + a|\) and \(|x - a|\) respectively. The electric field components from these charges are \(E_1 = \frac{kq}{(x+a)^2} \) (towards \(x = a\)) and \(E_2 = \frac{kq}{(x-a)^2} \) (away from \(x = -a\)). Adding these, the net electric field along the \(x\)-axis is: \[ E_x(x) = \frac{kq}{(x-a)^2} - \frac{kq}{(x+a)^2} \].

Analysis and Graphing

The function derived, \(E_x(x) = \frac{kq}{(x-a)^2} - \frac{kq}{(x+a)^2}\), is an even function, which means it's symmetric about the \(y\)-axis (x=0). Plotting \(E_x(x)\) for \(x\) ranging from \(-4a\) to \(4a\), we see spikes or rapid increases in the electric field near \(x = a\) and \(x = -a\). The field changes direction at the origin, resulting in a symmetric curve.

Key concepts

Coulomb's Law

Coulomb’s Law is a fundamental principle in electromagnetism that describes the force between two point charges. It is based on the formula: \( F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \), where \( F \) is the force between the charges, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, \( r \) is the distance between the charges, and \( k \) is Coulomb’s constant (approximately \( 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \) in a vacuum).
Understanding Coulomb’s Law helps us predict how charges interact:

• The force is attractive if the charges are opposite and repulsive if they are the same.
• Force magnitude is inversely proportional to the square of the distance between charges.
• For the exercise, at any point on the x-axis, the electric field is derived from this fundamental law.

This concept is valuable in calculating forces and fields in various configurations of charges, such as in the exercise where two point charges are symmetrically placed.

Point Charges

Point charges refer to idealized charges located at a single point in space. They provide a simplified model to analyze electrostatic problems with fewer complexities.
In our exercise, each charge \( q \) is located on the x-axis at positions \( x = a \) and \( x = -a \).

• Point charges generate an electric field in the surrounding space, described by Coulomb's Law.
• For any point along the x-axis, the direction of the field depends on the location of the charge and the point of measurement.
• The fields generated by these point charges are vectors, meaning they have both magnitude and direction.

When these charges are used in problem-solving, simplifying them as point charges allows us to focus purely on the charge interactions without worrying about their spatial structure.

Electric Field Graph

An electric field graph visually represents how the electric field intensity varies with position.
In our scenario, the graph plots the x-component of the electric field, \( E_x(x) \), along the x-axis. From \( -4a \) to \( 4a \), the function \( E_x(x) = \frac{kq}{(x-a)^2} - \frac{kq}{(x+a)^2} \) helps generate this graph.

• Axes: Horizontal corresponds to position (x-axis), vertical corresponds to the field \( E_x(x) \).
• The graph shows symmetry and includes spikes near \( x = a \) and \( x = -a \), due to proximity to charges.
• At \( x = 0 \), the electric field is zero, resulting in a crossing point on the graph.

This type of graph is crucial in visualizing how fields change with distance and understanding their directional nature around point charges.

Symmetric Charge Distribution

Symmetric charge distribution refers to a physical configuration where charges are evenly and mirror-imaged around a central point or axis.
In the current exercise, the point charges are precisely symmetric about the origin, since one is at \( x = a \) and the other at \( x = -a \).

• Resulting field at the center (\( x = 0 \)) is zero due to symmetry.
• For any point symmetrically positioned, the field contributions from each charge will have elements that cancel each other.
• This property simplifies calculations and predictions about the electric field at different points along the axis.
• The symmetry ensures that field variations are predictable and can be exploited in various applications of physics and engineering.

Grasping symmetric charge distribution is key to understanding how fields and forces arise in balanced arrangements and offers intuitive solutions to complex problems.