Find the electric field strength vector, if the potential of this field has the form φ = a → · r → , where a → is a constant vector and r → is the radius vector of a point of the field.

Given potential, φ = a → · r → where, a is constant. Let a → = a x i ^ + a y j ^ + a z k ^ and r → = x i ^ + y j ^ + z k ^ Then, φ = a → · r → = a x x + a y y + a z z Thus, electric field, in terms of potential is, E → = - ∇ → φ where, ∇ → = ∂ ∂ x i ^ + ∂ ∂ y j ^ + ∂ ∂ z k ^ E → = - ∂ ∂ x i ^ + ∂ ∂ y j ^ + ∂ ∂ z k ^ a x x + a y y + a z z E → = - ∂ ∂ x a x x i ^ + ∂ ∂ y a y y j ^ + ∂ ∂ z a z z k ^ = - a x i ^ + a y j ^ + z k ^ = - a →

Demonstrate that the potential of the field generated by a dipole, with electric moment P → , may be represented as φ = p → · r → 4 π ε 0 r 3 , where r → is the radius vector. Using this expression, find the magnitude of the electric field strength vector as a function of r and θ .

Assume that there are two charges, + q and - q , separated by a distance l . Then, the electric potential at a point, at a distance r > > l from this dipole, is φ r = + q 4 π ε 0 r + + - q 4 π ε 0 r - = q 4 π ε 0 r - - r + r + r - . . . 1 where, r - = Distance of point S from charge - q , r + = Distance of point S from charge + q . But, r - - r + = l cosθ and r + r - ≅ r 2 . . . 2 From equations 1 and 2 , φ r = q l cosθ 4 π ε 0 r 2 = p cosθ 4 π ε 0 r 2 φ = p → · r → 4 π ε 0 r 3 where, p is the magnitude of electric moment vector. Now, E r = - ∂ φ ∂ r = 2 p cosθ 4 π ε 0 r 3 and E θ = - ∂ φ r ∂ θ = p sinθ 4 π ε 0 r 3 So, E = E r 2 + E 0 2 = p 4 π ε 0 r 3 4 cos 2 θ + sin 2 θ

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There is an infinitely long straight thread carrying a charge with linear density $λ = 0.40 μC m^{- 1}$ . Calculate the potential difference between points $1$ and $2$ , if point $2$ is removed, $η = 2.0$ times farther from the thread than point $1$ .

Important Questions on ELECTRODYNAMICS

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT ELECTRIC FIELD IN VACUUM>Q 3.32.

Find the electric field potential and strength at the centre of a hemisphere of radius

R

, charged uniformly with the surface density

σ

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT ELECTRIC FIELD IN VACUUM>Q 3.33.

A very thin round plate of radius

R

carrying a uniform surface charge density

σ

is located in vacuum. Find the electric field potential and strength along the planet's axis as a function of a distance

l

from its centre. Investigate the obtained expression at

l \to 0

and

l ≫ R .

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT ELECTRIC FIELD IN VACUUM>Q 3.34.

Find the potential

φ

at the edge of a thin disc of radius

R

carrying the uniformly distributed charge with surface density

σ

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT ELECTRIC FIELD IN VACUUM>Q 3.35.

Find the electric field strength vector, if the potential of this field has the form

φ = \vec{a} \cdot \vec{r}

, where

\vec{a}

is a constant vector and

\vec{r}

is the radius vector of a point of the field.

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JEE Main

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT ELECTRIC FIELD IN VACUUM>Q 3.36.

Determine the electric field strength vector, if the potential of this field depends on $x$ and $y$ coordinates as,
(a) $φ = a (x^{2} - y^{2});$ (b) $φ = a x y,$

where $a$ is a constant. Draw the approximate shape of these fields using lines of force. (In the $x, y$ plane)

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JEE Main

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT ELECTRIC FIELD IN VACUUM>Q 3.37.

The potential of a certain electrostatic field has the form

φ = a (x^{2} + y^{2}) + b z^{2}

, where

a

and

b

are constants. Find the magnitude and direction of the electric field strength vector. What shape have the equipotential surfaces, in the following cases:
(a)

a > 0, b > 0;

(b)

a > 0, b < 0

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT ELECTRIC FIELD IN VACUUM>Q 3.38.

A charge

q

is uniformly distributed over the volume of a sphere of radius

R

. Assuming the permittivity to be equal to unity throughout, find the potential
(a) at the centre of the sphere;
(b) inside the sphere, as a function of the distance

r

from its centre.

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT ELECTRIC FIELD IN VACUUM>Q 3.39.

Demonstrate that the potential of the field generated by a dipole, with electric moment $\vec{P}$ , may be represented as $φ = \frac{\vec{p} \cdot \vec{r}}{4 π ε_{0} r^{3}}$ , where $\vec{r}$ is the radius vector. Using this expression, find the magnitude of the electric field strength vector as a function of $r$ and $θ$ .

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There is an infinitely long straight thread carrying a charge with linear density λ=0.40 μC m-1. Calculate the potential difference between points 1 and 2, if point 2 is removed, η=2.0 times farther from the thread than point 1.

Important Questions on ELECTRODYNAMICS

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There is an infinitely long straight thread carrying a charge with linear density $λ = 0.40 μC m^{- 1}$ . Calculate the potential difference between points $1$ and $2$ , if point $2$ is removed, $η = 2.0$ times farther from the thread than point $1$ .