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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

A point charge \(q\) is situated at a distance \(r\) from one end of a thin conducting rod of length \(\mathrm{L}\) having a charge \(\mathrm{Q}\) (uniformly distributed along its length). The magnitude of electric force between the two, is ...... \((\mathrm{A})[(2 \mathrm{kqQ}) / \mathrm{r}(\mathrm{r}+\mathrm{L})]\) (B) \([(\mathrm{kq} \mathrm{Q}) / \mathrm{r}(\mathrm{r}+\mathrm{L})]\) (C) \([(\mathrm{kqQ}) / \mathrm{r}(\mathrm{r}-\mathrm{L})]\) (D) \([(\mathrm{kQ}) / \mathrm{r}(\mathrm{r}+\mathrm{L})]\)

A simple pendulum of period \(\mathrm{T}\) has a metal bob which is negatively charged. If it is allowed to oscillate above a positively charged metal plate, its period will ...... (A) Remains equal to \(\mathrm{T}\) (B) Less than \(\mathrm{T}\) (C) Infinite (D) Greater than \(\mathrm{T}\)

Two point positive charges \(q\) each are placed at \((-a, 0)\) and \((a, 0)\). A third positive charge \(q_{0}\) is placed at \((0, y)\). For which value of \(\mathrm{y}\) the force at \(q_{0}\) is maximum \(\ldots \ldots \ldots\) (A) a (B) \(2 \mathrm{a}\) (C) \((\mathrm{a} / \sqrt{2})\) (D) \((\mathrm{a} / \sqrt{3})\)

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) \(\tan ^{-1}(\sqrt{3} / 2)\)

Two air capacitors \(A=1 \mu F, B=4 \mu F\) are connected in series with $35 \mathrm{~V}\( source. When a medium of dielectric constant \)\mathrm{K}=3$ is introduced between the plates of \(\mathrm{A}\), change on the capacitor changes by (A) \(16 \mu \mathrm{c}\) (B) \(32 \mu \mathrm{c}\) (C) \(28 \mu \mathrm{c}\) (D) \(60 \mu \mathrm{c}\)
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