Question

A positive point charge of magnitude \(q_{1}\) lies at the origin. Derive an expression for the incremental work done in moving a second point charge \(q_{2}\) through a distance \(d x\) from the starting position \((x, y, z)\), in the direction of \(-\mathbf{a}_{x}\)

Short answer

The expression for the incremental work done is given by: $$ d W = -k \frac{q_{1}q_{2}}{r^3} x d x $$

Step-by-step solution

Determine the force between the point charges

Using Coulomb's Law, we can determine the force exerted by the point charge \(q_{1}\) on the point charge \(q_{2}\). The magnitude of this force is given by: $$ F = k \frac{q_{1}q_{2}}{r^2} $$ where \(k\) is Coulomb's constant, and \(r\) is the distance between the charges, which can be found using the starting position, \((x, y, z)\). The distance is given by: $$ r = \sqrt{x^2 + y^2 + z^2} $$

Find the force vector

Next, we need to find the force vector acting on the charge \(q_{2}\). Since the force is acting along the line joining the two charges, we can find the unit vector along the position vector from the origin to the position \((x, y, z)\). The position vector, \(\mathbf{r}\), is given by: $$ \mathbf{r} = x\mathbf{a}_{x} + y\mathbf{a}_{y} + z\mathbf{a}_{z} $$ The unit vector along the position vector is then given by: $$ \mathbf{\hat{r}} = \frac{\mathbf{r}}{r} = \frac{x\mathbf{a}_{x} + y\mathbf{a}_{y} + z\mathbf{a}_{z}}{\sqrt{x^2 + y^2 + z^2}} $$ Now, the force vector \(\mathbf{F}\) acting on \(q_{2}\) is given by: $$ \mathbf{F} = F \mathbf{\hat{r}} $$

Find the work done by the force

The work done by the force \(\mathbf{F}\) when the charge \(q_{2}\) is moved through a distance \(d x\) in the direction of \(-\mathbf{a}_{x}\) can be found using the dot product of the force vector and the displacement vector. The displacement vector \(\mathbf{d r}\) is given by: $$ \mathbf{d r} = -d x\,\mathbf{a}_{x} $$ Now, the incremental work done, \(d W\), is given by: $$ d W = \mathbf{F} \cdot \mathbf{d r} $$

Calculate the incremental work done

Substitute the expressions for the force vector and displacement vector to find the incremental work done: $$ d W = (F \mathbf{\hat{r}}) \cdot (-d x\,\mathbf{a}_{x}) = (k \frac{q_{1}q_{2}}{r^2} \cdot \frac{x\mathbf{a}_{x} + y\mathbf{a}_{y} + z\mathbf{a}_{z}}{\sqrt{x^2 + y^2 + z^2}}) \cdot (-d x\,\mathbf{a}_{x}) $$ Since the dot product is only non-zero when the corresponding unit vectors match, we can simplify the expression to: $$ d W = -k \frac{q_{1}q_{2}}{r^3} x d x $$ This is the expression for the incremental work done in moving the second point charge through a distance \(d x\) in the direction of \(-\mathbf{a}_{x}\) from the starting position \((x, y, z)\).

Key concepts

Electric Force

Electric force is a fundamental concept in physics and plays a crucial role in understanding interactions between point charges. It is a force exerted by charged particles on each other. The magnitude of this force is determined using Coulomb's Law.

Coulomb’s Law states that the electric force \( F \) between two point charges can be calculated using the expression:

• \( F = k \frac{q_1 q_2}{r^2} \)

Here, \( k \) is Coulomb's constant, approximately \( 8.9875 \, \times \, 10^9 \, \text{Nm}^2/\text{C}^2 \), \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.

This law dictates that the force is attractive if the charges have opposite signs and repulsive if they have the same sign. The force is also directly proportional to the product of the charges, meaning larger charges exert more force on each other, and inversely proportional to the square of their separation, indicating that the force decreases as they move apart.

Remember, force not only has a magnitude but also a direction. In this scenario, the force acts along the line connecting the two charges.

Point Charge Interactions

Point charge interactions involve the study of forces and fields generated by point charges. These actions are central to understanding electrostatic principles.

When discussing point charges, the distance between them is crucial. In a three-dimensional space, this distance \( r \) can be found using the coordinates of the charges at positions \((x, y, z)\). Using the formula:

• \( r = \sqrt{x^2 + y^2 + z^2} \)

This expression helps us determine how far apart the charges are, allowing us to calculate the forces they exert on each other precisely using Coulomb's Law.

The force vector \( \mathbf{F} \) acting on a charge is derived from the magnitude of the force and the unit vector along the direction of the force. The unit vector \( \mathbf{\hat{r}} \) is given by:

• \( \mathbf{\hat{r}} = \frac{\mathbf{r}}{r} \)

Where \( \mathbf{r} = x\mathbf{a}_{x} + y\mathbf{a}_{y} + z\mathbf{a}_{z} \) is the position vector. This helps translate the force magnitude into a vectorial form, which is essential for calculating work done or analyzing the system's dynamics.

Work Done in Electric Fields

Work done in electric fields relates to moving a charge within an electric field, a fundamental process depicting energy changes in electrostatic systems.

The work \( dW \) done to move charge \( q_2 \) through displacement \( \mathbf{d r} \) is calculated using the dot product of the force vector \( \mathbf{F} \) and the displacement vector. In this scenario, the displacement vector is:

• \( \mathbf{d r} = -d x \, \mathbf{a}_{x} \)

The work done is then given by:

• \( dW = \mathbf{F} \cdot \mathbf{d r} \)

Applying the components of the force and direction, work done simplifies to an expression that considers only the component of force that aligns with the displacement. This yields an incremental work done:

• \( dW = -k \frac{q_{1}q_{2}}{r^3} x \, dx \)

This expression shows that work is done as energy is spent in moving charge \( q_2 \) through a slight distance \( dx \) against the electric field surrounding \( q_1 \), giving insight into the mechanical energy transfer in electric fields.