Question

An electric dipole is placed along the \(\mathrm{x}\) -axis at the origin o. \(\mathrm{A}\) point \(P\) is at a distance of \(20 \mathrm{~cm}\) from this origin such that OP makes an angle \((\pi / 3)\) with the x-axis. If the electric field at P makes an angle \(\theta\) with the x-axis, the value of \(\theta\) would be \(\ldots \ldots \ldots\) (A) \((\pi / 3)+\tan ^{-1}(\sqrt{3} / 2)\) (B) \((\pi / 3)\) (C) \((2 \pi / 3)\) (D) \(\tan ^{-1}(\sqrt{3} / 2)\)

Short answer

The value of θ would be \(\tan^{-1}(\sqrt{3})\). The correct answer is (D).

Step-by-step solution

Calculate the electric field components

To find the electric field components, we first need to rewrite the electric dipole in its Cartesian coordinates. We know that P is 20 cm from the origin at an angle of π/3. So, the coordinates of P are: P(x, y) = \((20 \cos(\pi / 3), 20 \sin(\pi / 3))\) Given that the electric field formula for a dipole along the x and y axes is: \(E_x = \frac{2 p_x}{4\pi\epsilon_0} \frac{x}{(x^2 + y^2)^{3/2}}\) \(E_y = \frac{p_x}{4\pi\epsilon_0} \frac{y}{(x^2 + y^2)^{3/2}}\) We can find the electric field along the x and y axes at point P using the electric dipole moment p_x: \(E_x = \frac{2 p_x}{4\pi\epsilon_0} \frac{(20\cos(\pi / 3))}{((20\cos(\pi / 3))^2 + (20\sin(\pi / 3))^2)^{3/2}}\) \(E_y = \frac{p_x}{4\pi\epsilon_0} \frac{(20\sin(\pi / 3))}{((20\cos(\pi / 3))^2 + (20\sin(\pi / 3))^2)^{3/2}}\)

Find the overall electric field at point P

The overall electric field at point P can be found by adding the electric field components along the x-axis (E_x) and the y-axis (E_y). This gives us: \(E_P = \sqrt{E_x^2 + E_y^2}\) Substituting the values for E_x and E_y, we get: \(E_P = \sqrt{ \frac{(2\cos(\pi / 3))^2p_x^2}{(4\pi\epsilon_0)^2(20^2\cos^2(\pi / 3) + 20^2\sin^2(\pi / 3))^2} + \frac{(\sin(\pi / 3))^2p_x^2}{(4\pi\epsilon_0)^2(20^2\cos^2(\pi / 3) + 20^2\sin^2(\pi / 3))^2}}\)

Calculate the angle θ

Now, we can find the angle θ, that the electric field at P makes with the x-axis by using the tangent function: \(\tan(\theta) = \frac{E_y}{E_x}\) So, the angle θ is given by: \(\theta = \tan^{-1} \left( \frac{E_y}{E_x} \right)\) Substituting the values for E_x and E_y, we get: \(\theta = \tan^{-1} \left( \frac{\frac{p_x}{4\pi\epsilon_0} \frac{(20\sin(\pi / 3))}{((20\cos(\pi / 3))^2 + (20\sin(\pi / 3))^2)^{3/2}}}{\frac{2 p_x}{4\pi\epsilon_0} \frac{(20\cos(\pi / 3))}{((20\cos(\pi / 3))^2 + (20\sin(\pi / 3))^2)^{3/2}}} \right)\) Simplifying this expression, we get: \(\theta = \tan^{-1} \left( \frac{\sin(\pi / 3)}{2\cos(\pi / 3)} \right) = \tan^{-1} \left( \frac{\sqrt{3} / 2}{2(1 / 2)} \right) = \tan^{-1}(\sqrt{3})\) Therefore, the value of θ would be \(\tan^{-1}(\sqrt{3})\). The correct answer is (D).

Key concepts

Electric Field Components

When dealing with electric dipoles, understanding the electric field components is crucial. An electric dipole consists of two equal charges of opposite sign separated by a distance.
This creates a unique electric field pattern.
The dipole's electric field can be split into two components: along the x-axis and along the y-axis.

• For a dipole, the component along the x-axis, denoted as \(E_x\), often involves the strength of the dipole moment and the distance from the dipole.
• Similarly, \(E_y\), the component along the y-axis, depends on the angle and position of the point of interest.

Calculating these components requires using the electric field formulas for a dipole. Plug the coordinates of the point into these formulas to find \(E_x\) and \(E_y\).
Each component reflects how the electric field stretches in respective directions due to the dipole's presence.

Cartesian Coordinates

In geometry and algebra, cartesian coordinates are vital for locating positions in a plane via two numbers.
In the context of electric dipoles, they help us specify the position of points related to the dipole.
To locate point \(P\) with respect to the dipole:

• Use the angle from the x-axis and the distance to convert into \(x\) and \(y\) coordinates.
• The formulae used are \(x = r \cos(\alpha)\) and \(y = r \sin(\alpha)\), where \(r\) is the distance and \(\alpha\) is the angle with the x-axis.

Cartesian coordinates simplify the vector calculations needed to determine electric field components from the dipole.

Angle Calculation

Calculating angles in physics problems often requires understanding relationships between different vectors.
For the dipole and electric field problem, determining the angle \(\theta\) that the field makes with the x-axis is a significant task.
The angle \(\theta\) signifies the field direction relative to a reference axis.

• First, find the tangent of \(\theta\) using \(\tan(\theta) = \frac{E_y}{E_x}\), where \(E_y\) and \(E_x\) are field components.
• Then, use the arctangent function \(\theta = \tan^{-1}(\frac{E_y}{E_x})\) to find \(\theta\).

Angle calculations provide insights into how much the electric field lines deviate from the reference axes, thus revealing the field's orientation.

Tangent Function

The tangent function connects an angle in a right triangle to the ratio of two specific sides.
In trigonometry, \(\tan(\theta) = \frac{opposite}{adjacent}\), linking angle calculation with geometry.
In electric field calculations, it helps relate the components of the field to the angle formed:

• The opposite side in our context is the electric field component aligned with the y-axis, \(E_y\).
• The adjacent side is the x-axis component, \(E_x\).
• Using these, \(\tan(\theta) = \frac{E_y}{E_x}\) gives the angle's tangent from field components.

Reversing this relationship with the \(\tan^{-1}\) function helps derive the angle value \(\theta\), giving the orientation of the electric field in relation to the coordinate axes.