SATHEE: Electrostatics 4 Question 1

Electrostatics 4 Question 1

1. A point dipole $p = - p_{0} \hat{x}$ is kept at the origin. The potential and electric field due to this dipole on the $Y$ -axis at a distance $d$ are, respectively [Take, $V = 0$ at infinity]

(a) $\frac{| p |}{4 π ε_{0} d^{2}}, \frac{p}{4 π ε_{0} d^{3}}$

(b) $0, \frac{- p}{4 π ε_{0} d^{3}}$

(d) $\frac{| p |}{4 π ε_{0} d^{2}}, \frac{- p}{4 π ε_{0} d^{3}}$

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Answer:

Correct Answer: 1. (b)

Solution:

The given problem can be shown as clearly potential difference at point $P$ due to dipole is

$\begin{aligned} V & = V_{A P} + V_{B P} (scalar addition) \\ \Rightarrow V & = \frac{k (- q)}{A P} + \frac{k (q)}{B P} \end{aligned}$

Here, $A P = B P = \sqrt{a^{2} + r^{2}}$

$∴ V = - \frac{k q}{\sqrt{a^{2} + r^{2}}} + \frac{k q}{\sqrt{a^{2} + r^{2}}} = 0$

Now, electric field at any point on $Y$ -axis, i.e. equatorial line of the dipole can be given by

$\begin{aligned} E = - \frac{k p}{r^{3}} \\ \Rightarrow & E = - \frac{1}{4 π ε_{0}} \frac{p}{r^{3}} \\ Given, & r & = d \\ ∴ & E & = - \frac{1}{4 π ε_{0}} \frac{p}{d^{3}} \end{aligned}$

From Eqs. (ii) and (iii), correct option is (b).

Alternate Solution

Electric field at any point at $θ$ angle from axial line of dipole is given by

$E = - \frac{k p}{r^{3}} \sqrt{3 \cos^{2} θ + 1}$

Here, $θ = 90^{\circ} \Rightarrow \cos θ = \cos 90^{\circ} = 0$ and $r = d$

$∴ E = - \frac{k p}{d^{3}} = - \frac{p}{4 π ε_{0} d^{3}}$

Electrostatics 4 Question 1

1. A point dipole $p = - p_{0} \hat{x}$ is kept at the origin. The potential and electric field due to this dipole on the $Y$ -axis at a distance $d$ are, respectively [Take, $V = 0$ at infinity]

Answer:

Solution:

Alternate Solution

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Electrostatics 4 Question 1

1. A point dipole p=−p0x^ is kept at the origin. The potential and electric field due to this dipole on the Y-axis at a distance d are, respectively [Take, V=0 at infinity]

Answer:

Solution:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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1. A point dipole $p = - p_{0} \hat{x}$ is kept at the origin. The potential and electric field due to this dipole on the $Y$ -axis at a distance $d$ are, respectively [Take, $V = 0$ at infinity]