Electric Field and Charge Particle in Electric Field

Q 11 :

A small bob of mass 100 mg and charge +10 $μ$ C is connected to an insulating string of length 1 m. It is brought near to an infinitely long non-conducting sheet of charge density ' $σ$ ' as shown in figure. If string subtends an angle of 45° with the sheet at equilibrium the charge density of sheet will be:

(Given, $ε_{0} = 8.85 \times 10^{- 12} \frac{F}{m}$ and acceleration due to gravity, $g = 10 m / s^{2}$ ) [2025]

0.885 $nC / m^{2}$
17.7 $nC / m^{2}$
885 $nC / m^{2}$
1.77 $nC / m^{2}$

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(4)

T sin 45° = Eq

T cos 45° = mg

$Eq = mg$

$E = \frac{m g}{q} = \frac{σ}{2 ε_{0}}$

$σ = \frac{2 ε_{0} m g}{q} = 1.77 nC / m^{2}$

Q 12 :

Consider a circular loop that is uniformly charged and has a radius $a \sqrt{2}$ . Find the position along the positive z-axis of the cartesian coordinate system where the electric field is maximum if the ring was assumed to be placed in xy-plane at the origin: [2025]

$\frac{a}{\sqrt{2}}$
$\frac{a}{2}$
a
0

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(3)

$E = K \frac{Q r}{{(x^{2} + R^{2})}^{3 / 2}}$

For $E_{\max}, \frac{d E}{d x} = 0$

$∴ x = \frac{R}{\sqrt{2}} = \frac{\sqrt{2} a}{\sqrt{2}} = a$

Q 13 :

A metallic ring is uniformly charged as shown in figure. AC and BD are two mutually perpendicular diameters. Electric field due to arc AB to 'O' is 'E' in magnitude. What would be the magnitude of electric field at 'O' due to arc ABC? [2025]

2 E
$\sqrt{2} E$
E/2
Zero

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(2)

Q 14 :

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : The outer body of an air craft is made of metal which protects persons sitting inside from lightning strikes.

Reason (R) : The electric field inside the cavity enclosed by a conductor is zero.

In the light of the above statements, choose the most appropriate answer from the options given below: [2025]

Both (A) and (R) are correct and (R) is the correct explanation of (A)
(A) is correct but (R) is not correct
Both (A) and (R) are correct but (R) is not the correct explanation of (A)
(A) is not correct but (R) is correct

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(1)

The outer body of aircraft is made with metal to provide electrostatic shielding such that electric field inside craft remain zero for any charge on outer surface.

Q 15 :

Electric charge is transferred to an irregular metallic disk as shown in figure. If $σ_{1}, σ_{2}, σ_{3}$ and $σ_{4}$ are charge densities at given points then, choose the correct answer from the options given below: [2025]

(A) $σ_{1} > σ_{3}; σ_{2} = σ_{4}$

(B) $σ_{1} > σ_{2}; σ_{3} > σ_{4}$

(C) $σ_{1} > σ_{3} > σ_{2} = σ_{4}$

(D) $σ_{1} > σ_{2}; σ_{3} = σ_{4}$

(E) $σ_{1} = σ_{2} = σ_{3} = σ_{4}$

A, B and C Only
A and C Only
D and E Only
B and C Only

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(1)

$σ \propto \frac{1}{R} R \to radius of curvature$

$R_{2} = R_{4} > R_{3} > R_{1}$

$\Rightarrow σ_{1} > σ_{3} > σ_{4} = σ_{2}$

Q 16 :

Electric field in a certain region is given by $\vec{E} = (\frac{A}{x^{2}} \hat{i} + \frac{B}{y^{3}} \hat{j}) .$ The SI unit of $A$ and $B$ are [2023]

${Nm}^{3} C^{- 1}; {Nm}^{2} C^{- 1}$
${Nm}^{2} C^{- 1}; {Nm}^{3} C^{- 1}$
${Nm}^{3} C; {Nm}^{2} C$
${Nm}^{2} C; {Nm}^{3} C$

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(2)

$\vec{E} = \frac{A}{x^{2}} \hat{i} + \frac{B}{y^{3}} \hat{j}$

$[\frac{A}{x^{2}}] = {NC}^{- 1} \Rightarrow [A] = {Nm}^{2} C^{- 1}$

$[\frac{B}{y^{3}}] = {NC}^{- 1} \Rightarrow [B] = {Nm}^{3} C^{- 1}$

Q 17 :

Graphical variation of electric field due to a uniformly charged insulating solid sphere of radius $R$ , with distance $r$ from the centre $O$ is represented by [2023]

[IMAGE 128]

[IMAGE 129]
[IMAGE 130]
[IMAGE 131]
[IMAGE 132]

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(4)

$Electric field of solid sphere (uniformly charged)$

$E (r)$ ${\begin{matrix} \frac{Q}{4 π ε_{0} r^{2}}, & r \geq R \\ \frac{Q r}{4 π ε_{0} R^{3}}, & r \leq R \end{matrix}$

$Graphically, E (r) \propto r for r \leq R$

Q 18 :

A stream of positively charged particles having $\frac{q}{m} = 2 \times 10^{11} \frac{C}{kg}$ and velocity ${\vec{v}}_{0} = 3 \times 10^{7} \hat{i} m/s$ deflected by an electric field $1.8 \hat{j} kV/m .$ The electric field exists in a region of 10 cm along the $x$ -direction. Due to the electric field, the deflection of the charged particles in the $y$ -direction is ________ mm. [2023]

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(2)

[IMAGE 133]

$a = \frac{F}{m} = \frac{q E}{m} = (2 \times 10^{11}) (1.8 \times 10^{3}) = 3.6 \times 10^{14} {m/s}^{2}$

$Time to cross plates t = \frac{d}{v}$

$t = \frac{0.10}{3 \times 10^{7}}$

$y = \frac{1}{2} a t^{2} = \frac{1}{2} (3.6 \times 10^{14}) {(\frac{0.01}{9 \times 10^{14}})}^{2}$

$= 0.2 \times 0.01 = 0.002 m = 2 mm$

Q 19 :

A uniform electric field of 10 N/C is created between two parallel charged plates (as shown in fig). An electron enters the field symmetrically between the plates with a kinetic energy 0.5 eV. The length of each plate is 10 cm. The angle $(θ)$ of deviation of the path of electron as it comes out of the field is ________ (in degree). [2023]

[IMAGE 134]

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(45)

$l = u t$

$t = \frac{l}{u}$

$\tan θ = \frac{v_{y}}{u} = \frac{a t}{u} = \frac{E q}{m} \frac{l}{u^{2}} = \frac{E q l}{2 (\frac{1}{2} m u^{2})} = \frac{E q l}{2 K.E.}$

$\tan θ = \frac{10 \times 1.6 \times 10^{- 19} \times 0.1}{2 \times 0.5 \times 1.6 \times 10^{- 19}} = 1 \Rightarrow θ = 45 °$

Q 20 :

Two equal positive point charges are separated by a distance $2 a$ . The distance of a point from the centre of the line joining the two charges on the equatorial line (perpendicular bisector), at which the force experienced by a test charge $q_{0}$ becomes maximum, is $\frac{a}{\sqrt{x}} .$ The value of $x$ is __________ . [2023]

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(2)

[IMAGE 135]

$F = \frac{2 K q q_{0} x}{{(x^{2} + a^{2})}^{3 / 2}}$

$For F to be maximum, \frac{d F}{d x} = 0$

$\Rightarrow x = \frac{a}{\sqrt{2}}$

Topic Question Set

Q 12 :

Consider a circular loop that is uniformly charged and has a radius $a \sqrt{2}$ . Find the position along the positive z-axis of the cartesian coordinate system where the electric field is maximum if the ring was assumed to be placed in xy-plane at the origin: [2025]

Q 13 :

A metallic ring is uniformly charged as shown in figure. AC and BD are two mutually perpendicular diameters. Electric field due to arc AB to 'O' is 'E' in magnitude. What would be the magnitude of electric field at 'O' due to arc ABC? [2025]

Q 16 :

Electric field in a certain region is given by $\vec{E} = (\frac{A}{x^{2}} \hat{i} + \frac{B}{y^{3}} \hat{j}) .$ The SI unit of $A$ and $B$ are [2023]

Q 17 :

Graphical variation of electric field due to a uniformly charged insulating solid sphere of radius $R$ , with distance $r$ from the centre $O$ is represented by [2023]

[IMAGE 128]

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