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Chapter 11: Problem 1558

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

Expert verified
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

01

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||.
02

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: \(\dfrac{2kp\cos(\theta)}{r^3} = \dfrac{kp\sin(\theta)}{r^3}\)
03

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: \(\theta = \tan^{-1}(\sqrt{2})\)
04

Compare the result to given options

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

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Most popular questions from this chapter

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