Question

At what angle \(\theta\) a point \(P\) must be located from dipole axis so that the electric field intensity at the point is perpendicular to the dipole axis? (A) \(\tan ^{-1}(1 / \sqrt{2})\) (B) \(\tan ^{-1}(1 / 2)\) (C) \(\tan ^{-1}(2)\) (C) \(\tan ^{-1}(\sqrt{2})\)

Short answer

The angle θ at which a point P must be located from the dipole axis so that the electric field intensity at the point is perpendicular to the dipole axis is (C) \(\tan^{-1}(\sqrt{2})\).

Step-by-step solution

Calculate the components of electric field

Due to the dipole, the electric field at point P can be divided into two components. One along the dipole axis (E||) and the other perpendicular to the dipole axis (E⊥). For an electric field to be perpendicular to the axis, it must satisfy the condition E⊥=E||.

Use electric field formula to write the components

The components of the electric field E⊥ and E|| produced by an electric dipole can be given by: \(E_{\parallel} = \dfrac{2kp\cos(\theta)}{r^3}\) and \(E_\perp = \dfrac{kp\sin(\theta)}{r^3}\) where k is the electrostatic constant, p is the dipole moment, r is the distance from the centre of the dipole to the point P, and θ is the angle we have to find. Since E⊥=E||, we can set up the following equation: \(\dfrac{2kp\cos(\theta)}{r^3} = \dfrac{kp\sin(\theta)}{r^3}\)

Solve the equation for θ

We can simplify the equation and solve for θ: \(\dfrac{\sin(\theta)}{\cos(\theta)} = \dfrac{2\cos(\theta)}{\sin(\theta)}\) Cross-multiplying, we get: \(\sin^2(\theta) = 2\cos^2(\theta)\) Recall that \(sin^2(\theta) = 1 - cos^2(\theta)\), so substituting we get: \(1 - \cos^2(\theta) = 2\cos^2(\theta)\) Next, we have: \(3\cos^2(\theta) = 1\) Taking the square root on both sides and remembering that cos(θ) = 1 / √3 for θ in the first quadrant, we end up with: \(\cos(\theta) = \dfrac{1}{\sqrt{3}}\) Now we can find θ using the inverse cosine function: \(\theta = \cos^{-1}(\dfrac{1}{\sqrt{3}})\) This is equivalent to: \(\theta = \tan^{-1}(\sqrt{2})\)

Compare the result to given options

Comparing our result with the given options, we can see that the correct answer is: (C) \(\tan^{-1}(\sqrt{2})\)

Key concepts

electric field components

When dealing with an electric dipole, it's important to understand that the electric field at any point can be described by two components. This helps to analyze how the field interacts in different directions relative to the dipole.

• The parallel component, denoted as \(E_{||}\), aligns along the dipole axis. It shows how the field behaves when moving in the same direction as the dipole itself.
• The perpendicular component, \(E_{\perp}\), acts at a right angle to the dipole axis. It displays the field's behavior in the direction that's perpendicular to the dipole.

The combination of these two components provides a complete picture of the electric field at any point due to the dipole. For any problem involving an electric dipole, identifying these components helps us to determine how the electric field behaves under various conditions. Remember that these components are critical in understanding the electrostatic interactions in a dipole system.

angle with dipole axis

The angle \(\theta\) between a point and the dipole axis plays a crucial role in defining how the electric field components behave. This angle is key to determining the direction and magnitude of the electric field.

• It influences the relative strength of the perpendicular and parallel components of the electric field at a given point.
• Changes in \(\theta\) can dramatically affect how these components relate to one another, especially when certain conditions need to be satisfied.

In our problem, the goal was to find the specific angle \(\theta\) at which the electric field is perpendicular to the dipole axis, implying that the two field components have a defined relationship. Understanding this angle is crucial because it tells us how the forces are distributed around the dipole, which is important for practical applications in physics and engineering.

electric field perpendicularity

Electric field perpendicularity occurs when the electric field vector forms a right angle with the dipole axis. This special condition simplifies the analysis of the electric field because it balances the parallel and perpendicular components.

• For this to happen, the magnitudes of the components must satisfy the relation \(E_{||} = E_{\perp}\). This means the field's influence along the axis equals its effect perpendicular to it.
• When analyzing problems involving perpendicularity, the equations describing each component allow us to find the precise angle \(\theta\) where this condition holds.

The concept of electric field perpendicularity is useful in many areas of physics because it represents a point of equilibrium between different directional forces around a dipole, often making calculations more straightforward.

dipole moment

The dipole moment is a fundamental property of a dipole that measures the strength and direction of its electric field. It is given by the product of the charge and the separation between the charges.

• Symbolized as \(p\), it plays a critical role in determining the behavior of the electric field components around a dipole.
• The dipole moment directly influences the magnitudes of \(E_{||}\) and \(E_{\perp}\), impacting how the electric field spreads in space.

In our examination of the given problem, the dipole moment aids in forming the mathematical relationships needed to find the desired angle where an electric field is perpendicular. A higher dipole moment means a stronger electric field, which can affect the angle \(\theta\) more significantly. Understanding the dipole moment is vital for predicting how a dipole's electric field behaves in different environments.