Question

An electric dipole (dipole moment $\vec{p} = p \, \hat{i}$), consisting of charges $-q$ and $q$, separated by distance $2a$, is placed along the $x$-axis, with its center at the origin. Show that the potential $V$, due to this dipole, at a point $x$ ($x \gg a$) is equal to: \[ V = \frac{1}{4\pi\epsilon_0} \cdot \frac{\vec{p} \cdot \hat{i}}{x^2}. \]

Answer 1

(i) The potential \( V \) due to two point charges \( +q \) and \( -q \) separated by a distance \( 2a \) can be calculated using: \[ V = \frac{1}{4\pi \epsilon_0} \left( \frac{q}{x - a} - \frac{q}{x + a} \right) \] Simplifying this, we get: \[ V = \frac{q}{4\pi \epsilon_0} \left( \frac{1}{x - a} - \frac{1}{x + a} \right) \] The common denominator simplifies to: \[ V = \frac{q}{4\pi \epsilon_0} \cdot \frac{(x + a) - (x - a)}{x^2 - a^2} \] Thus, the potential becomes: \[ V = \frac{q}{4\pi \epsilon_0} \cdot \frac{2a}{x^2 - a^2} \] Now, since \( p = 2a q \) is the dipole moment, the potential can be written as: \[ V = \frac{p}{4\pi \epsilon_0} \cdot \frac{1}{x^2 - a^2} \] If \( x \gg a \), the potential simplifies to: \[ V \approx \frac{p}{4\pi \epsilon_0} \cdot \frac{1}{x^2} \] Alternatively, we can use the geometry of the situation to derive the following expression for the potential: \[ V = \frac{1}{4\pi \epsilon_0} \left( \frac{q}{r_1} - \frac{q}{r_2} \right) \] where \( r_1 \) and \( r_2 \) are the distances from the charges to the point where the potential is being calculated. Using geometry, we can express \( r_1 \) and \( r_2 \) as: \[ r_1 = \sqrt{r^2 - 2a r \cos \theta} \] \[ r_2 = \sqrt{r^2 + 2a r \cos \theta} \] Now using the binomial expansion and retaining terms up to the first order in \( a/r \), we get: \[ \frac{1}{r_1} \approx \frac{1}{r} \left( 1 + \frac{a \cos \theta}{r} \right) \] \[ \frac{1}{r_2} \approx \frac{1}{r} \left( 1 - \frac{a \cos \theta}{r} \right) \] Thus, the potential \( V \) can be expressed as: \[ V = \frac{q}{4\pi \epsilon_0} \cdot \frac{2a \cos \theta}{r^2} \] Since \( \mathbf{p} \cdot \hat{r} = p \cos \theta \), we can write: \[ V = \frac{1}{4\pi \epsilon_0} \cdot \frac{p}{r^2} \] This is the expression for the potential due to an electric dipole.