Two charged concentric spherical shells have radii 10 . 0 cm and 15 . 0 cm . The charge on the inner shell is 7 . 50 × 10 - 8 C , and that on the outer shell is 6 . 33 × 10 - 8 C . Find the electric field (a) at r = 12 . 0 cm and (b) at r = 20 . 0 cm .

According to the problem, we draw the above figure. Given: Charge on inner sphere, Q 1 = 7 . 50 × 10 - 8 C Charge on outer sphere, Q 2 = 6 . 33 × 10 - 8 C Radius of inner sphere, R 1 = 10 . 0 cm = 10 . 0 × 10 - 2 m Radius of outer sphere, R 2 = 15 . 0 cm = 15 . 0 × 10 - 2 m (a) Given, O A = r = 12 . 0 cm = 12 × 10 - 2 m Electric field at point A is given by E A = K Q 1 O A 2 = 9 × 10 9 × 7 . 50 × 10 - 8 12 × 10 - 2 2 = 46885 N C - 1 = 4 . 68 × 10 4 N C - 1 . (b) Given, O B = r = 20 . 0 cm = 20 × 10 - 2 m Electric field at r > R 2 at point B is given by E B = K Q 1 O B 2 + K Q 2 O B 2 = K O B 2 Q 1 + Q 2 ⇒ E B = 9 × 10 9 20 × 10 - 2 2 7 . 50 × 10 - 8 + 6 . 33 × 10 - 8 ⇒ E B = 3 . 17 × 10 9 N C - 1 .

Two charged concentric spherical shells have radii 10 . 0 cm and 15 . 0 cm . The charge on the inner shell is 7 . 50 × 10 - 8 C , and that on the outer shell is 6 . 33 × 10 - 8 C . Find the electric field (a) at r = 12 . 0 cm and (b) at r = 20 . 0 cm .

According to the problem, we draw the above figure. Given: Charge on inner sphere, Q 1 = 7 . 50 × 10 - 8 C Charge on outer sphere, Q 2 = 6 . 33 × 10 - 8 C Radius of inner sphere, R 1 = 10 . 0 cm = 10 . 0 × 10 - 2 m Radius of outer sphere, R 2 = 15 . 0 cm = 15 . 0 × 10 - 2 m (a) Given, O A = r = 12 . 0 cm = 12 × 10 - 2 m Electric field at point A is given by E A = K Q 1 O A 2 = 9 × 10 9 × 7 . 50 × 10 - 8 12 × 10 - 2 2 = 46885 N C - 1 = 4 . 68 × 10 4 N C - 1 . (b) Given, O B = r = 20 . 0 cm = 20 × 10 - 2 m Electric field at r > R 2 at point B is given by E B = K Q 1 O B 2 + K Q 2 O B 2 = K O B 2 Q 1 + Q 2 ⇒ E B = 9 × 10 9 20 × 10 - 2 2 7 . 50 × 10 - 8 + 6 . 33 × 10 - 8 ⇒ E B = 3 . 17 × 10 9 N C - 1 .

Two long, charged, thin-walled, concentric cylindrical shells have radii of 3 . 0 cm and 6 . 0 cm . The charge per unit length is - 7 . 0 × 10 - 6 C m - 1 on the inner shell and + 5 . 0 × 10 - 6 C m - 1 on the outer shell. What is the (a) magnitude E and (b) direction (radially inward or outward) of the electric field at radial distance r = 4 . 0 cm ? What is (c) E and (d) the direction at r = 8 . 0 cm ?

Given: Radius of inner shell, R 1 = 3 . 0 cm = 3 . 0 × 10 - 2 m . Radius of outer shell, R 2 = 6 . 0 cm = 6 × 10 - 2 m . Charge per unit length on inner shell, λ 1 = - 7 . 0 × 10 - 6 C m - 1 . Charge per unit length on outer shell, λ 2 = + 5 . 0 × 10 - 6 C m - 1 . (a) Electric field at r = 4 . 0 cm (which is R 1 < r < R 2 ), E = 2 K λ 1 r = 2 × 9 × 10 9 × - 7 . 0 × 10 - 6 4 . 0 × 10 - 2 E = - 1 . 84 × 10 6 N C - 1 . Magnitude of electric field is 1 . 84 × 10 6 N C - 1 . (b) This electric field is directed inwards as inner shell is negatively charged. (c) Electric field at r = 8 . 0 cm = 8 . 0 × 10 - 2 m , r > R 2 > R 1 E = 2 K λ 1 r + 2 K λ 2 r = 2 K r λ 1 + λ 2 = 2 × 9 × 10 9 8 × 10 - 2 - 7 × 10 - 6 + 5 × 10 - 6 = - 4 . 5 × 10 5 N C - 1 . Magnitude of electric field is 4 . 5 × 10 5 N C - 1 . (d) Electric field is negative, which means it is directed radially inwards.

An electron is shot directly toward the centre of a large metal plate that has surface charge density - 1 . 50 × 10 - 6 C / m 2 . If the initial kinetic energy of the electron is 3 . 93 × 10 - 17 J and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be?

Given : Surface charge density = 1 . 50 × 10 6 C m - 2 Initial kinetic energy of the electron = - 3 . 93 × 10 - 17 J Here total kinetic energy is lost by electron and got converted or transformed into electrostatic potential energy We know that ∆ U = q ∆ V = - q ∫ E → . d r → Using work-energy theorem, W All forces = k f - k i , where k f is the final kinetic energy and k i is the initial kinetic energy. W e = k f - k i As we have very large sheet which can be considered as infinite sheet metal sheet, the electric field is given by E → = σ ε 0 i ^ From work-energy theorem W All forces = k f - k i W c = k f - k i ∆ W c = - ∆ U - e E . dx = 0 - k i - e σ ε 0 0 - x = - k i + 1 . 6 × 10 - 19 × 1 . 5 × 10 - 6 × x 8 . 854 × 10 - 12 = + 3 . 93 × 10 - 17 x = 3 . 93 × 10 - 17 × 8 . 854 × 10 - 12 1 . 6 × 10 - 19 × 1 . 5 × 10 - 6 on solving x = 1 . 449 × 10 - 3 m x = 0 . 001449 m = 1 . 449 mm ≃ 1 . 45 mm

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Two charged concentric spherical shells have radii $10.0 cm$ and $15.0 cm$ . The charge on the inner shell is $7.50 \times 10^{- 8} C$ , and that on the outer shell is $6.33 \times 10^{- 8} C$ . Find the electric field (a) at $r = 12.0 cm$ and (b) at $r = 20.0 cm$ .

Important Questions on Gauss' Law

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Principles Of Physics International Student Version>Chapter 23 - Gauss' Law>Problems>Q 26

An electron is released

9.0 cm

from a very long nonconducting rod with a uniform linear charge density

4.5 μC m^{- 1}

. What is the magnitude of the electron's initial acceleration?

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Principles Of Physics International Student Version>Chapter 23 - Gauss' Law>Problems>Q 27

Two long, charged, thin-walled, concentric cylindrical shells have radii of

3.0 cm

and

6.0 cm

. The charge per unit length is

- 7.0 \times 10^{- 6} C m^{- 1}

on the inner shell and

+ 5.0 \times 10^{- 6} C m^{- 1}

on the outer shell. What is the (a) magnitude

E

and (b) direction (radially inward or outward) of the electric field at radial distance

r = 4.0 cm

? What is (c)

E

and (d) the direction at

r = 8.0 cm

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Principles Of Physics International Student Version>Chapter 23 - Gauss' Law>Problems>Q 28

The figure shows a spherical shell with uniform volume charge density $ρ = 1.56 nC m^{- 3}$ , inner radius $a = 10.0 cm$ , and outer radius $b = 2.00 a$ . What is the magnitude of the electric field at radial distances (a) $r = 0$ , (b) $r = a / 2.00$ , (c) $r = a$ , (d) $r = 1.50 a$ , (e) $r = b$ and (f) $r = 3.00 b$ ?

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Principles Of Physics International Student Version>Chapter 23 - Gauss' Law>Problems>Q 29

In the figure shown below, a solid sphere of radius $a = 2.00 cm$ is concentric with a spherical conducting shell of inner radius $b = 2.00 a$ and outer radius $c = 2.40 a$ . The sphere has a net uniform charge $q_{1} = + 2.00 fC$ ; the shell has a net charge $q_{2} = - q_{1}$ . What is the magnitude of the electric field at radial distances (a) $r = 0,$ (b) $r = \frac{a}{2.00},$ (c) $r = a$ (d) $r = 1.50 a$ (e) $r = 2.30 a,$ and (f) $r = 3.50 a ?$ What is the net charge on the (g) inner and (h) outer surface of the shell?

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Principles Of Physics International Student Version>Chapter 23 - Gauss' Law>Problems>Q 30

In the fig. a butterfly net is in a uniform electric field of magnitude $E = 4.5 mN C^{- 1}$ . The rim, a circle of radius $a = 11 cm$ is aligned perpendicular to the field. The net contains no net charge. Find the electric flux through the net.

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Principles Of Physics International Student Version>Chapter 23 - Gauss' Law>Problems>Q 31

A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitude

E = K r^{4}

, directed radially outward from the center of the sphere. Here

r

is the radial distance from that center, and

K

is a constant. What is the volume density

ρ

of the charge distribution?

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Principles Of Physics International Student Version>Chapter 23 - Gauss' Law>Problems>Q 32

Figure (a) shows a narrow charged solid cylinder that is coaxial with a larger charged cylindrical shell. Both are non-conducting and thin and have uniform surface charge densities on their outer surfaces. Figure (b) gives the radial component $E$ of the electric field versus radial distance $r$ from the common axis, and $E_{s} = 6.0 \times 10^{3} N C^{- 1}$ What is the shell's linear charge density?

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$(a) (b)$

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Principles Of Physics International Student Version>Chapter 23 - Gauss' Law>Problems>Q 33

An electron is shot directly toward the centre of a large metal plate that has surface charge density

- 1.50 \times 10^{- 6} C / m^{2} .

If the initial kinetic energy of the electron is

3.93 \times 10^{- 17} J

and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be?

Two charged concentric spherical shells have radii 10.0 cm and 15.0 cm. The charge on the inner shell is 7.50×10-8 C, and that on the outer shell is 6.33×10-8 C. Find the electric field (a) at r=12.0 cm and (b) at r=20.0 cm.

Important Questions on Gauss' Law

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Two charged concentric spherical shells have radii $10.0 cm$ and $15.0 cm$ . The charge on the inner shell is $7.50 \times 10^{- 8} C$ , and that on the outer shell is $6.33 \times 10^{- 8} C$ . Find the electric field (a) at $r = 12.0 cm$ and (b) at $r = 20.0 cm$ .