Electric Charges and Fields Test

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Electric Charges and Fields Test - 22

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Question 1
1 / -0

A hollow metal sphere of radius R is uniformly charged. The electric field due to the sphere at a distance r from the centre

A
Increases as r increases for $$r < R$$ and for $$ r > R$$

B
Zero as r increases for $$ r < R$$, decreases as r increases for $$r > R $$

C
Zero as r increases for $$r < R$$, increases as r increases for $$r > R$$

D
Decreases as r increases for $$ r < R$$ and for $$r > R $$

Solution

Charge Q will be distributed over the surface of hollow metal sphere.
(i) For r < R (inside)
By Gauss law, $$\oint \bar{E_{in}} . \overline{dS} = \dfrac{q_{en}}{\varepsilon_0}= 0$$
$$\Rightarrow E_{in} = 0$$ $$(\because q_{en} = 0)$$
(ii) For r > R (out side)
$$\oint \bar{E}_0 . \overline{dS} = \dfrac{q_{en}}{\varepsilon_0}$$
Here, $$q_{en} = Q (\because q_{en} = Q)$$
$$\therefore E_0 4 \pi r^2 = \dfrac{Q}{\varepsilon _0}$$
$$\therefore E_0 \propto \dfrac{1}{r^2}$$
Question 2
1 / -0

Electric field outside a long wire carrying charge q is proportional to :

A
$$\dfrac { 1 }{ r } $$

B
$$\dfrac { 1 }{ { r }^{ 2 } } $$

C
$$\dfrac { 1 }{ { r }^{ \frac { 1 }{ 3/5 } } } $$

D
$$\dfrac { 1 }{ { r }^{ \frac { 1 }{ 3/2 } } } $$

Solution

Using Gauss Law across the cylindrical Gaussian surface,
$$\displaystyle\int E.dS = \dfrac{q_{enc}}{\epsilon_0}$$
$$\therefore E\times 2\pi r \times l = \dfrac{\lambda\times l}{\epsilon_0}$$
$$\therefore E = \dfrac{\lambda}{2\pi r\epsilon_0}$$
Here $$\lambda$$ is the linear charge density on the wire.
Question 3
1 / -0

An electric line of force in the $$xy-plane$$ is given by equation $$x^{2} + y^{2} = 1$$. A particle with unit positive charge, initially at rest at the point $$x = 1, y = 0$$ in the $$xy - plane$$.

A
Not move at all

B
Will move along straight line

C
Will move along the circular line of force

D
Information is insufficient to draw any conclusion

Solution

Charge will move along the circular line of force because $$x^{2} + y^{2} = 1$$ is the equation of circle in $$xy - plane$$.
Question 4
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Two identical charged spheres suspended from a common distance $$d(d < < 1)$$ apart because of their mutual repulsion. The charged begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity $$v$$. Then $$v$$ varies as a function of the distance $$x$$ between the spheres, as

A
$$v\propto x^{\dfrac {1}{2}}$$

B
$$v\propto x$$

C
$$v\propto x^{-\dfrac {1}{2}}$$

D
$$v\propto x^{-1}$$

Solution

From figure $$\tan \theta = \dfrac {F_{e}}{mg} =\theta$$
$$\dfrac {kq^{2}}{x^{2}mg} = \dfrac {x}{2l}$$
or $$x^{3}\propto q^{2} .... (1)$$
or $$x^{3/2} \propto q ..... (2)$$
Differentiating eq. $$(1)$$ w.r.t time
$$3x^{2} \dfrac {dx}{dt}\propto 2q \dfrac {dq}{dt}$$ but $$\dfrac {dq}{dt}$$ is constant
so $$x^{2}(v) \propto q$$ Replace $$q$$ from eq. $$(2)$$
$$x^{2}(v) \propto x^{3/2}$$ or $$v\propto x^{-1/2}$$.
Question 5
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Two unlike charges attract each other with a force of 10N. If the distance between them is doubled, the force between them is :

A
$$40N$$

B
$$20N$$

C
$$5N$$

D
$$2.5N$$

Solution

We know $$F=\dfrac {1}{4\pi \varepsilon_0}\dfrac {q_1q_2}{r^2}=10 N$$

if distance is doubled.
$$F_{new}=\dfrac {1}{4\pi \varepsilon_0}\cdot \dfrac {q_1q_2}{(2r)^2}=\dfrac {10}{4}$$

$$=2\cdot 5 N$$
Question 6
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Number of electric lines of force emanating from $$1C$$ of positive charge in vacuum is :

A
$$8.85\times10 ^{-12}$$

B
$$9\times10 ^{9}$$

C
$$1/4\pi \times 9\times 10^{9}$$

D
$$1.13\times10 ^{11}$$

Solution

$$\phi $$ from $$1C $$ of charge is $$= \dfrac{Q_{enclosed}}{\epsilon_o}$$
$$= \dfrac{1}{8.8501\times 10^{-12}}$$

$$= 1.129\times 10^{11}$$
Question 7
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A force ‘F’ is acting between two charges in air. If the space between them be completely filled with a medium $$K=4$$, the force will be :

A
$$F$$

B
$$4F$$

C
$$\dfrac{F}{4}$$

D
$$2F$$

Solution

By Coulomb's law, the force between two charges $$(q_1, q_2)$$ in air is given by $$F=\dfrac{q_1q_2}{4\pi\epsilon_0 r^2}$$
where $$r=$$ distance between two charges.
Now the force between two charges $$(q_1, q_2)$$ in medium with dielectric constant $$K=4$$ becomes, $$F'=\dfrac{q_1q_2}{4\pi K\epsilon_0 r^2}$$
So, $$F'=\dfrac{F}{K}=\dfrac{F}{4}$$
Question 8
1 / -0

An electron revolves around the nucleus of hydrogen atom in a circle of radius $$5\times 10^{-11}\ m$$. The intensity of electric field at a point in the orbit of the electron is:

A
$$5.76\times 10^{11}N/C$$

B
$$9.216\times 10^{-8}N/C$$

C
$$0$$

D
$$4N/C$$

Solution

We know hydrogen atom consist of one proton and one electron
$$R=5\times 10^{-11} m$$
We know attractive force $$=\dfrac {k q_1(q_2)}{r^2}=\dfrac {9\times 10^9\times (1\cdot 6\times 10^{-19})^2}{(5\times 10^{-11})^2}$$
We know $$F = E q$$
$$\therefore E=\dfrac {F}{q}$$
$$\therefore E=\dfrac {9\times 10^9\times (1\cdot 6\times 10^{-19})}{5\times 10^{-11})^2}=5\cdot 76\times 10^{11} N/C.$$
Question 9
1 / -0

A charge $$q_1$$ exerts a force of $$45N$$ on a charge of $$q_{2}=10^{-5} C$$ located at a point $$0.2m$$ from $$q_{1}$$. The magnitude of $$q_{1}$$ is

A
$$4\times 10^{-8} C$$

B
$$2\times 10^{-5} C$$

C
$$3\times 10^{-6} C$$

D
$$5\times 10^{-8} C$$

Solution

As we know electrostatic force between two charges $$F=\dfrac {k q_1 q_2}{r^2}$$

$$\Rightarrow 45=\dfrac {(9\times 10^9)(q_1)(10^{-5})}{(0\cdot 2)^2}$$

$$\Rightarrow q_1=\dfrac {45\times 0\cdot 2\times 0\cdot 2}{9\times 10^9\times 10^{-5}}$$

$$\Rightarrow q_1=2\times 10^{-5}C$$
Question 10
1 / -0

When we remove polyester or woollen clothes in dark, we can see a spark and hear a crackling sound. Which of the following is responsible for it?

A
Static electricity.

B
Current electricity.

C
Reflection of light.

D
Refraction of light.

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Electric Charges and Fields Test - 22

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Electric Charges and Fields Test - 22

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Question 1

Increases as r increases for $$r < R$$ and for $$ r > R$$

Zero as r increases for $$ r < R$$, decreases as r increases for $$r > R $$

Zero as r increases for $$r < R$$, increases as r increases for $$r > R$$

Decreases as r increases for $$ r < R$$ and for $$r > R $$

Solution

Question 2

$$\dfrac { 1 }{ r } $$

$$\dfrac { 1 }{ { r }^{ 2 } } $$

$$\dfrac { 1 }{ { r }^{ \frac { 1 }{ 3/5 } } } $$

$$\dfrac { 1 }{ { r }^{ \frac { 1 }{ 3/2 } } } $$

Solution

Question 3

Not move at all

Will move along straight line

Will move along the circular line of force

Information is insufficient to draw any conclusion

Solution

Question 4

$$v\propto x^{\dfrac {1}{2}}$$

$$v\propto x$$

$$v\propto x^{-\dfrac {1}{2}}$$

$$v\propto x^{-1}$$

Solution

Question 5

$$40N$$

$$20N$$

$$5N$$

$$2.5N$$

Solution

Question 6

$$8.85\times10 ^{-12}$$

$$9\times10 ^{9}$$

$$1/4\pi \times 9\times 10^{9}$$

$$1.13\times10 ^{11}$$

Solution

Question 7

$$F$$

$$4F$$

$$\dfrac{F}{4}$$

$$2F$$

Solution

Question 8

$$5.76\times 10^{11}N/C$$

$$9.216\times 10^{-8}N/C$$

$$0$$

$$4N/C$$

Solution

Question 9

$$4\times 10^{-8} C$$

$$2\times 10^{-5} C$$

$$3\times 10^{-6} C$$

$$5\times 10^{-8} C$$

Solution

Question 10

Static electricity.

Current electricity.

Reflection of light.

Refraction of light.

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