Question

Let a wire occupying the line segment from \((0,-c)\) to \((0, c)\) in the \(x y\)-plane have a constant charge density equal to \(\rho\). Show that the electrostatic potential due to this wire at a point \(\left(x_{1}, y_{1}\right)\) of the \(x y\)-plane is given by $$ \Phi=\rho \log \frac{\sqrt{x_{1}^{2}+\left(c-y_{1}\right)^{2}}+c-y_{1}}{\sqrt{x_{1}^{2}+\left(c+y_{1}\right)^{2}}-c-y_{1}}+k $$ where \(k\) is an arbitrary constant. Show that if \(k\) is chosen so that \(\Phi(1,0)=0\), then, as \(c\) becomes infinite, \(\Phi\) approaches the limiting value \(-2 \rho \log \left|x_{1}\right|\). This is the potential of an infinite wire with uniform charge.

Short answer

Answer: The electrostatic potential at a point (x₁, y₁) due to an infinite wire with constant charge density ρ along the y-axis is given by Φ = -2ρ log|x₁|.

Step-by-step solution

Setup the Equation

Given the potential due to a line of charge formula, we can write the equation to find the potential at the point \((x_1, y_1)\): $$ \Phi(x_1, y_1) = \frac{\rho}{4\pi\epsilon_0} \int_{-c}^{c} \frac{1}{\sqrt{x_1^2+(y-y_1)^2}} dy $$

Perform the Integration

We will integrate the expression inside the integral: $$ \Phi(x_1, y_1) = \frac{\rho}{4\pi\epsilon_0} \left[ \frac{(y-y_1)}{\left|x_1\right|} \arctan \frac{(y-y_1)}{\left|x_1\right|} \right]_{-c}^{c} $$

Evaluate the Result and Include the Arbitrary Constant

Now, we will obtain the final expression for the potential by substituting the limits of integration and including the arbitrary constant \(k\): $$ \Phi(x_1, y_1) = \rho \log \frac{\sqrt{x_1^{2}+\left(c-y_1\right)^{2}}+c-y_1}{\sqrt{x_1^{2}+\left(c+y_1\right)^{2}}-c-y_1} + k $$

Determine the Value of k

We will now determine the constant \(k\) by setting \(\Phi(1, 0) = 0\). Substituting these values into the equation and solving for \(k\), we get: $$ k = -\rho \log \frac{\sqrt{1+\left(c\right)^{2}}+c}{\sqrt{1+\left(c\right)^{2}}-c} $$

Find the Limiting Value as c Approaches Infinity

Finally, we will find the limiting value of the potential as \(c\) approaches infinity: $$ \lim_{c\to\infty} \Phi = \rho \log \frac{\sqrt{x_1^{2}+\left(y_1\right)^{2}}}{\left|x_1\right|} - 2\rho \log \left|x_1\right| = -2 \rho \log \left|x_1\right| $$ This is the potential of an infinite wire with uniform charge.

Key concepts

Charge Density

Charge density, represented by the symbol \(\rho\), is a measure of how much electric charge is distributed over a particular length, surface, or volume. For a line of charge like a wire, the term linear charge density is used and is defined as the charge per unit length of the wire. In our exercise, we consider a constant linear charge density, meaning that the charge is distributed evenly along the segment of the wire from \( (0, -c) \) to \( (0, c) \). Understanding charge density is crucial when calculating the electrostatic potential created by charged objects because it significantly influences the electric field in the surrounding space.

When dealing with problems like the one presented, charge density serves as a starting point for setting up the expression that will be integrated to find the potential, because the electric field at any point in space is directly related to the charge distribution that creates it.

Electrostatic Potential

Electrostatic potential, denoted as \(\Phi\), is a scalar quantity that represents the amount of work needed to move a unit positive charge from a reference point (usually infinity) to a specific point in the field without any acceleration. It is influenced by the spatial configuration of the charge distribution and can be calculated by integrating the electric field along a path. In the context of our exercise, the electrostatic potential at a point \( (x_1, y_1) \) in the plane due to a charged wire is being calculated. It's important to grasp that the potential is a function of the position in the plane relative to the wire and encapsulates the effect of the entire charge distribution, not just the nearest point on the wire.

The key advantage of working with electrostatic potential is that it is a scalar quantity – unlike the electric field which is a vector – making certain calculations simpler, such as those involving energy or work.

Line of Charge

A line of charge is a theoretical simplification used in physics to model a very long object with charge distributed along its length. This model is especially handy when the object in question is much longer than it is wide, such as a thin wire, allowing for a one-dimensional analysis of the electric effects it creates. In our example, the wire occupies a finite segment on the y-axis and has a uniform charge density, thus qualifying as a 'line of charge.'

Understanding the concept of a line of charge is critical for predicting the behavior of electric fields and potentials around such objects. When we deal with infinite lines of charge, the electrostatic potential tends to simplify, reaching a form that depends only on the distance from the line of charge and not on the other coordinates, due to the symmetry and infinite extent of the distribution.

Integration Calculus

Integration is an essential mathematical tool in calculus used for finding the areas under curves, among other applications. In physics, and specifically in electromagnetism, integration helps determine quantities like electric field and electrostatic potential when the system has continuous charges. When calculating the potential due to a continuous charge distribution as presented in our exercise, we integrate the contribution of each infinitesimal element of charge over the entire distribution.

Integration calculus becomes indispensable when the charge distribution is not point-like but rather spread out, such as with a line of charge. It enables us to accumulate all the small contributions to the potential from each segment of the wire. The integral takes into account both the amount of charge and its varying distances from the point where the potential is being calculated, leading to an accurate and complete description of the potential in the field created by the charge distribution.