Electric Field and Charge Particle in Electric Field

Q 1 :
An infinitely long positively charged straight thread has a linear charge density $λ$ $C m^{- 1}$ . An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is [2024]

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(2)

Electric field E at a distance r due to infinite long wire is $E = \frac{2 k λ}{r}$

Force of electron, $F = e E \Rightarrow F = e (\frac{2 k λ}{r}) = \frac{2 k λ e}{r}$

This force will provide the required centripetal force.

$F = \frac{m v^{2}}{r} = \frac{2 k λ e}{r} \Rightarrow v = \sqrt{\frac{2 k λ e}{m}}$

$K E = \frac{1}{2} m v^{2} = \frac{1}{2} m (\frac{2 k λ e}{m}) = k λ e = constant$

Q 2 :
$σ$ is the uniform surface charge density of a thin spherical shell of radius R. The electric field at any point on the surface of the spherical shell is: [2024]

$\frac{σ}{ε_{0} R}$
$\frac{σ}{ε_{0}}$
$\frac{σ}{2 ε_{0}}$
$\frac{σ}{4 ε_{0}}$

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(2)

By Gauss law $\int \vec{E} \cdot d \vec{A} = \frac{q_{in}}{ε_{0}}$

$E d A = \frac{σ \times d A}{ε_{0}} \Rightarrow E = \frac{σ}{ε_{0}}$

Q 3 :
A particle of charge ‘−q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘ $+ λ$ ’. Then time period will be given as

(Consider k as Coulomb’s constant) [2024]

$T^{2} = \frac{4 π^{2} m}{2 k λ q} r^{3}$
$T = 2 π r \sqrt{\frac{m}{2 k λ q}}$
$T = \frac{1}{2 π r} \sqrt{\frac{m}{2 k λ q}}$
$T = \frac{1}{2 π} \sqrt{\frac{2 k λ q}{m}}$

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(2)

$F_{e} = \frac{m v^{2}}{r} \Rightarrow q E = \frac{m v^{2}}{r}$

$q (\frac{2 K λ}{r}) = \frac{m v^{2}}{r} \Rightarrow v = \sqrt{\frac{2 K q λ}{m}}$

$T = \frac{2 π r}{v} = \frac{2 π r}{\sqrt{\frac{2 K q λ}{m}}} = 2 π r \sqrt{\frac{m}{2 K q λ}}$

Q 4 :
Two charges q and 3q are separated by a distance ‘r’ in air. At a distance x from charge q, the resultant electric field is zero. The value of x is [2024]

$\frac{r}{(1 + \sqrt{3})}$
$\frac{(1 + \sqrt{3})}{r}$
$r (1 + \sqrt{3})$
$\frac{r}{3 (1 + \sqrt{3})}$

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(1)

${({\vec{E}}_{net})}_{P} = 0 \Rightarrow \frac{k q}{x^{2}} = \frac{k \cdot 3 q}{{(r - x)}^{2}}$

${(r - x)}^{2} = 3 x^{2}$

$r - x = \sqrt{3} x \Rightarrow x = \frac{r}{\sqrt{3} + 1}$

Q 5 :
If the net electric field at point P along Y axis is zero, then the ratio of $| \frac{q_{2}}{q_{3}} |$ is $\frac{8}{5 \sqrt{x}}$ , where $x$ = ______ . [2024]

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(5)

$\frac{K q_{2}}{{(\sqrt{20})}^{2}} \cos β = \frac{K q_{3}}{{(5)}^{2}} \cos θ$

$\frac{K q_{2}}{20} \frac{4}{\sqrt{20}} = \frac{K q_{3}}{25} \frac{4}{\sqrt{25}}$

$\frac{q_{2}}{q_{3}} = \frac{20}{25} \frac{\sqrt{20}}{\sqrt{25}} = \frac{8}{5 \sqrt{x}}$

$\Rightarrow \sqrt{x} = \frac{8 \times 25 \sqrt{25}}{5 \times 20 \sqrt{20}} \Rightarrow x = 5$

Q 6 :
An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density $+ σ$ . The electron at t = 0 is at a distance of 1 m from S and has a speed of 1 m/s. The maximum value of $σ$ if the electron strikes S at t = 1s is $α [\frac{m ε_{0}}{e}] \frac{C}{m^{2}},$ the value of $α$ is _________ . [2024]

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(8)

Initially the electron must move away from sheet

Electric field due to infinite plane sheet of charge,

$E = \frac{σ}{2 ε_{0}}$

Force experienced by the electron towards the sheet, $F = e E$

Hence, acceleration of the electron, $a = \frac{F}{m}$

$a = \frac{e E}{m} = \frac{e σ}{2 ε_{0} m}$

Using $s = u t + \frac{1}{2} a t^{2}$

$- 1 = 1 \times 1 + \frac{1}{2} (- \frac{e σ}{2 ε_{0} m}) 1^{2}$

$\frac{e σ}{4 ε_{0} m} = 2 \Rightarrow σ = 8 \frac{ε_{0} m}{e}$

Q 7 :
Suppose a uniformly charged wall provides a uniform electric field of $2 \times 10^{4}$ N/C normally. A charged particle of mass 2g being suspended through a silk thread of length 20 cm and remains stayed at a distance of 10 cm from the wall. Then the charge on the particle will be $\frac{1}{\sqrt{x}} μ C$ where $x =$ __________. [use g = 10 $m / s^{2}$ ] [2024]

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(3)

$\sin θ = \frac{10}{20} = \frac{1}{2} \Rightarrow θ = 30 °$

Along x-axis

$\sum {(f_{net})}_{y} = 0 (for equilibrium)$

$T \sin θ = q E$ ...(i)

Along y-axis

$\sum {(f_{net})}_{x} = 0 (for equilibrium)$

$T \cos θ = m g$ ...(ii)

From (i) divide by (ii), $\tan θ = \frac{q E}{m g}$

$q = \frac{m g \tan θ}{E}$

$= \frac{2 \times 10^{- 3} \times 10}{2 \times 10^{4}} \times \frac{1}{\sqrt{3}} = \frac{10^{- 6}}{\sqrt{3}} C$

$\Rightarrow q = \frac{1}{\sqrt{3}} μ C \Rightarrow x = 3$

Topic Question Set

Q 2 :
$σ$ is the uniform surface charge density of a thin spherical shell of radius R. The electric field at any point on the surface of the spherical shell is: [2024]

Q 3 :
A particle of charge ‘−q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘ $+ λ$ ’. Then time period will be given as

(Consider k as Coulomb’s constant) [2024]

Q 4 :
Two charges q and 3q are separated by a distance ‘r’ in air. At a distance x from charge q, the resultant electric field is zero. The value of x is [2024]

Q 5 :
If the net electric field at point P along Y axis is zero, then the ratio of $| \frac{q_{2}}{q_{3}} |$ is $\frac{8}{5 \sqrt{x}}$ , where $x$ = ______ . [2024]

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Topic Question Set

Q 2 : σ is the uniform surface charge density of a thin spherical shell of radius R. The electric field at any point on the surface of the spherical shell is: [2024]

Q 3 : A particle of charge ‘−q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘+λ’. Then time period will be given as (Consider k as Coulomb’s constant) [2024]

Q 4 : Two charges q and 3q are separated by a distance ‘r’ in air. At a distance x from charge q, the resultant electric field is zero. The value of x is [2024]

Q 5 : If the net electric field at point P along Y axis is zero, then the ratio of |q2q3| is 85x, where x = ______ . [2024]

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Q 2 :
$σ$ is the uniform surface charge density of a thin spherical shell of radius R. The electric field at any point on the surface of the spherical shell is: [2024]

Q 3 :
A particle of charge ‘−q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘ $+ λ$ ’. Then time period will be given as

(Consider k as Coulomb’s constant) [2024]

Q 4 :
Two charges q and 3q are separated by a distance ‘r’ in air. At a distance x from charge q, the resultant electric field is zero. The value of x is [2024]

Q 5 :
If the net electric field at point P along Y axis is zero, then the ratio of $| \frac{q_{2}}{q_{3}} |$ is $\frac{8}{5 \sqrt{x}}$ , where $x$ = ______ . [2024]