Question

Two point charges of equal magnitude \(q\) are positioned at \(z=\pm d / 2 .(a)\) Find the electric field everywhere on the \(z\) axis; \((b)\) find the electric field everywhere on the \(x\) axis; \((c)\) repeat parts \((a)\) and \((b)\) if the charge at \(z=-d / 2\) is \(-q\) instead of \(+q\).

Short answer

Question: Find the net electric field along the z and x axes for two point charges with equal magnitudes in the first case, and one positive and one negative charge in the second case. Answer: Case 1 (Both charges have magnitude q): - Net electric field on z-axis: \(\vec{E}_z = \vec{E}_1 + \vec{E}_2\). - Net electric field on x-axis: \(\vec{E}_x = \vec{E}_1 + \vec{E}_2\). Case 2 (One charge has magnitude +q, and the other has magnitude -q): - Steps 1-3 from Case 1 should be repeated using the new charge magnitude for the second charge. The results will be the net electric fields along the z and x axes for this case.

Step-by-step solution

Define the position of the charges on the z-axis

We have two point charges of equal magnitude \(q\), located at \(z = \pm\frac{d}{2}\).

Calculate the electric field due to each charge

Let's assume a point P on the z-axis at a distance z from the origin. The distance between the charge at \(z = \frac{d}{2}\) and point P is \(r_1 = \left| z - \frac{d}{2} \right|\). Similarly, the distance between the charge at \(z = -\frac{d}{2}\) and point P is \(r_2 = \left| z + \frac{d}{2} \right|\). The electric field due to each charge at point P is given by: \(\vec{E}_1 = \frac{kq}{r_1^2}\hat{r}_1\) \(\vec{E}_2 = \frac{kq}{r_2^2}\hat{r}_2\)

Calculate the net electric field on the z-axis

Now we find the net electric field at point P by adding the fields due to both charges: \(\vec{E}_z = \vec{E}_1 + \vec{E}_2\) This gives us the net electric field everywhere on the z-axis. #b) Electric field on the x-axis#

Define the position of the charges on the x-axis

We now want to find the electric field on the x-axis. This means the position of point P is now given by \((x, 0, 0)\).

Calculate the electric field due to each charge along the x-axis

The distances from each charge to point P along the x-axis are: \(r_1 = \sqrt{x^2 + \left(\frac{d}{2}\right)^2}\) \(r_2 = \sqrt{x^2 + \left(-\frac{d}{2}\right)^2}\) The electric fields due to each charge are given by: \(\vec{E}_1 = \frac{kq}{r_1^2}\hat{r}_1\) \(\vec{E}_2 = \frac{kq}{r_2^2}\hat{r}_2\)

Calculate the net electric field on the x-axis

Finally, we find the net electric field on the x-axis by adding the fields due to both charges: \(\vec{E}_x = \vec{E}_1 + \vec{E}_2\) This gives us the net electric field everywhere on the x-axis. ## Case 2: One charge has magnitude +q, and the other has magnitude -q ## #c) Electric field on the z and x-axis# In this case, the electric fields due to each charge will have the same expressions as in Case 1, but the second charge will have a magnitude of \(-q\) instead of \(+q\). We can repeat Steps 1-3 for both parts (a) and (b) using this new charge magnitude to calculate the net electric field along the z and x axes.

Key concepts

Electrostatics

Electrostatics is the branch of physics that deals with the study of forces, fields, and potentials arising from static charges. It involves concepts such as charge, Coulomb's law, electric field, and potential.

To dive a bit deeper, charges are the basic units of electricity and can be either positive or negative. Like charges repel each other, while opposite charges attract. This is described by Coulomb's law, which quantifies the electrostatic force between two point charges. The law states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

An electric field represents the effect of a charge on the surrounding space. It exerts a force on other charges within the field. The field's strength at any point is the force experienced by a charge placed at that point, and its direction is the direction of the force on a positive test charge.

Electric Field Calculation

The calculation of the electric field generated by a point charge involves using Coulomb's law, which provides a simple formula:

\[\begin{equation}\vec{E} = \frac{kq}{r^2}\hat{r}\end{equation}\]
Here, \( \vec{E} \) is the electric field vector, \( k \) is Coulomb's constant, \( q \) is the charge of the point source, \( r \) is the distance from the point charge to the point of interest, and \( \hat{r} \) is the unit vector indicating the direction of the field. To find the total electric field at a particular point due to multiple charges, we need to calculate the vector sum of the individual fields generated by each charge. This process can become more complex when dealing with multiple point charges, as you need to consider the contribution of each charge separately and then combine them to get the total field.

For example, the charge at \( z = +\frac{d}{2} \) and the charge at \( z = -\frac{d}{2} \) each create their own electric field at any point in space. By separately calculating these fields and then adding them vectorially, we find the net electric field at any point.

Superposition Principle

The superposition principle is a fundamental concept in physics. It states that when multiple forces are acting at a point, the total force is simply the vector sum of these individual forces. This principle applies perfectly to electrostatics and electric fields.

For electric fields, this means the net electric field \( \vec{E}_{net} \) at any point is the vector sum of the electric fields \( \vec{E}_1, \vec{E}_2, \ldots, \vec{E}_n \) produced by each charge:

\[\begin{equation}\vec{E}_{net} = \vec{E}_1 + \vec{E}_2 + \ldots + \vec{E}_n\end{equation}\]
Such linear superposition is essential in determining the resultant field when two or more point charges are present. It allows us to analyze complex configurations by breaking them down into simpler components for which the electric fields can be calculated and then combined. As highlighted in our textbook exercise, when dealing with two point charges of equal magnitude, we first find the electric fields due to each charge independently and then apply the superposition principle to get the total electric field along different axes.