A solid sphere of radius R is uniformly charged with charge density ρ in its volume. A spherical cavity of radius R 2 is made in the sphere as shown in the figure. Find the electric potential at the centre of the sphere.

We know that electric potential at a point can be found using superposition principle.Given body can be assumed as a solid charged sphere having radius R and charge density ρ and a solid charged surface having radius R 2 and charge density - ρ placed at the position of the cavity. Net potential at the centre of bigger sphere (point C ) is sum of potential of bigger charged sphere at its centre and potential of smaller charged sphere at point C located on its periphery. V = V 1 + V 2 V = 3 k Q 2 R + k Q ' R 2 V = 3 k ρ × 4 3 π R 3 2 R + k - ρ × 4 3 π R 2 3 R 2 V = 2 π k ρ R 2 - 1 3 π k ρ R 2 V = 5 3 π k ρ R 2 = 5 ρ R 2 12 ε 0

A charge of 16 × 10 – 9 C is fixed at the origin of coordinates. A second charge of unknown magnitude is at x = 3 m , y = 0 and a third charge of 12 × 10 – 9 C is at x = 6 m , y = 0 . What is the value of the unknown charge if the resultant field at x = 8 m , y = 0 is 20 . 25 N C - 1 directed towards positive x - axis ?

Given, Net field at x = 8 , y = 0 E → net = 20 . 25 N C - 1 along x - axis . Suppose charge kept at x = 3 , y = 0 has magnitude q . Electric field due to a charge q 1 at a distance r is given by, E = k q 1 r 2 And if multiple charges are present, then from superposition principle we can say that net electric field at a point is vector sum of all the electric fields due to individual charges. E → = E → 1 + E → 2 + E → 3 + . . . . . . . . . . At x = 8 , y = 0 , net electric field due to all the charges, E → net = K 16 × 10 - 9 8 2 i ^ + K q 8 2 i ^ + K 12 × 10 - 9 8 2 i ^ 20 . 25 i ^ = 9 × 10 9 16 × 10 - 9 8 2 i ^ + 9 × 10 9 q 5 2 i ^ + 9 × 10 9 12 × 10 - 9 2 2 i ^ ( K = 9 × 10 9 N m 2 C - 2 ) 2 . 25 + 9 × 10 9 q 5 2 + 27 = 20 . 25 9 × 10 9 q 5 2 = - 9 q = - 25 × 10 9 C

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A long cylindrical shell carries positive surface charge $σ$ in the upper half and negative surface charge $- σ$ in the lower half. The electric field lines around the cylinder will look like figure given in (figures are schematic and not drawn to scale)

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Important Questions on Electrostatics

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Alpha Question Bank for Engineering: Physics>Chapter 22 - Electrostatics>Exercise - 3>Q 52

A uniformly charged solid sphere of radius

R

has potential

V_{0}

(measured with respect to

\infty

) on its surface. For this sphere the equipotential surfaces with potentials

\frac{3 V_{0}}{2}, \frac{5 V_{0}}{4}, \frac{3 V_{0}}{4} and \frac{V_{0}}{4}

have radius

R_{1}, R_{2}, R_{3}

and

R_{4}

, respectively. Then

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Alpha Question Bank for Engineering: Physics>Chapter 22 - Electrostatics>Exercise - 3>Q 53

The region between two concentric spheres of radii $a$ and $b$ , respectively (see figure), has volume charge density $ρ = \frac{A}{r}$ , where $A$ is a constant and $r$ is the distance from the centre. At the centre of the spheres is a point charge $Q$ . The value of $A$ such that the electric field in the region between the spheres will be constant, is

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Alpha Question Bank for Engineering: Physics>Chapter 22 - Electrostatics>Exercise - 3>Q 54

An electric dipole has a fixed dipole moment

\overset{⇀}{p}

, which makes angle

θ

with respect to

x -

axis. When subjected to an electric field

{\vec{E}}_{1} = E \hat{i}

, it experiences a torque

{\overset{⇀}{T}}_{1} = τ \hat{k}

. When subjected to another electric field

E_{2} = \sqrt{3} E_{1} \hat{j}

it experiences a torque

T_{2} = - T_{1}

. The angle

θ

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Alpha Question Bank for Engineering: Physics>Chapter 22 - Electrostatics>Exercise - 3>Q 55

Three concentric metal shells

A, B and C

of respective radii

a, b and c (a < b < c)

have surface charge densities

+ σ, - σ and + σ

, respectively. The potential of shell

B

is,

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Alpha Question Bank for Engineering: Physics>Chapter 22 - Electrostatics>Exercise - 4>Q 3

A solid sphere of radius $R$ is uniformly charged with charge density $ρ$ in its volume. A spherical cavity of radius $\frac{R}{2}$ is made in the sphere as shown in the figure. Find the electric potential at the centre of the sphere.

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Alpha Question Bank for Engineering: Physics>Chapter 22 - Electrostatics>Exercise - 4>Q 4

A uniform surface charge of density $σ$ is given to a quarter of a disc extending up to infinity in the first quadrant of $x - y$ plane. The centre of the disc is at the origin $O$ . Find the $z -$ component of the electric field at the point $(0, 0, z)$ and the potential difference between the points $(0, 0, d) & (0, 0, 2 d) .$

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Alpha Question Bank for Engineering: Physics>Chapter 22 - Electrostatics>Exercise - 4>Q 5

A charged ball of mass

5.88 \times 10^{- 4} kg

is suspended from two silk strings of equal lengths so that the strings are inclined at

90 °

with each other. Another ball carrying a charge which is equal in magnitude but opposite in sign of the first one is placed vertically below the first one at a distance of

4.2 \times 10^{- 2} m .

Due to this, the tension in the strings is doubled. Determine the charge on the ball and the tension in the strings after electrostatic interaction.

(g = 9.8 m s^{- 2})

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Alpha Question Bank for Engineering: Physics>Chapter 22 - Electrostatics>Exercise - 4>Q 6

A charge of

16 \times 10^{- 9} C

is fixed at the origin of coordinates. A second charge of unknown magnitude is at

x = 3 m, y = 0

and a third charge of

12 \times 10^{- 9} C

is at

x = 6 m, y = 0 .

What is the value of the unknown charge if the resultant field at

x = 8 m, y = 0

20.25 N C^{- 1}

directed towards positive

x - axis

A long cylindrical shell carries positive surface charge σ in the upper half and negative surface charge -σ in the lower half. The electric field lines around the cylinder will look like figure given in (figures are schematic and not drawn to scale)

Important Questions on Electrostatics

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A long cylindrical shell carries positive surface charge $σ$ in the upper half and negative surface charge $- σ$ in the lower half. The electric field lines around the cylinder will look like figure given in (figures are schematic and not drawn to scale)