A small bob of mass 100 mg and charge $+10 \text{ }\mu C$ is connected to an insulating string of length 1 m. It is brought near to an infinitely long non-conducting sheet of charge density '$\sigma$' as shown in figure. If string subtends an angle of $45^\circ$ with the sheet at equilibrium the charge density of sheet will be.
(Given, $\epsilon_0 = 8.85\times10^{-12} \frac{F}{m}$ and acceleration due to gravity, $g=10 \frac{m}{s^2}$)
$1.77 \text{ nC/m}^2$
To determine the charge density \(\sigma\) on the sheet, we analyze the forces acting on the charged bob in equilibrium.
Given:
- Mass of the bob, \(m = 100 \, \text{mg} = 100 \times 10^{-6} \, \text{kg}\)
- Charge on the bob, \(q = +10 \, \mu\text{C} = 10 \times 10^{-6} \, \text{C}\)
- Permittivity of free space, \(\epsilon_0 = 8.85\times10^{-12} \text{ F/m}\)
- Acceleration due to gravity, \(g = 10 \, \text{m/s}^2\)
In equilibrium, the forces acting on the bob are:
- Gravitational force, \(F_g = m \cdot g = 100 \times 10^{-6} \times 10 = 1 \times 10^{-3} \, \text{N}\)
- Electric force due to the sheet, \(F_e\)
- Tension in the string, which has components balancing both forces.
The tension's horizontal component provides the equilibrium condition for the electric force:
- \(T \sin(45^\circ) = F_e\)
- \(T \cos(45^\circ) = F_g\)
From these, we know:
- \(T = \frac{F_g}{\cos(45^\circ)}\)
- By substituting the value of \(T\) in the first equation: \(\frac{F_g}{\cos(45^\circ)} \cdot \sin(45^\circ) = F_e\)
- The electric field due to an infinite sheet is \(E = \frac{\sigma}{2\epsilon_0}\)
- The electric force \(F_e = q \cdot E = q \cdot \frac{\sigma}{2\epsilon_0}\)
From \(F_e = \frac{F_g \cdot \sin(45^\circ)}{\cos(45^\circ)}\), we solve for \(\sigma\):
\(q \cdot \frac{\sigma}{2\epsilon_0} = \frac{F_g \cdot \sin(45^\circ)}{\cos(45^\circ)}\)
Solving, we have:
\(\sigma = \frac{2 \cdot F_g \cdot \epsilon_0}{q} = \frac{2 \cdot 1 \times 10^{-3} \cdot 8.85 \times 10^{-12}}{10 \times 10^{-6}}\)
Calculating gives:
\(\sigma \approx 1.77 \, \text{nC/m}^2\)
Thus, the correct answer is 1.77 nC/m2.
Similar Questions
Two blocks of masses $m$ and $M$, ($M > m$), are placed on a frictionless table as shown in figure. A massless spring with spring constant $k$ is attached with the lower block. If the system is slightly displaced and released, then
A. The time period of small oscillation of the two blocks is $T = 2\pi \sqrt{\frac{(m+M)}{k}}$
B. The acceleration of the blocks is $a = \frac{kx}{M+m}$ (x = displacement of the blocks from the mean position)
C. The magnitude of the frictional force on the upper block is $\frac{\mu m|x|}{M+m}$
D. The maximum amplitude of the upper block, if it does not slip, is $\frac{\mu (M+m)g}{k}$
E. Maximum frictional force can be $\mu(M + m) g$.
Choose the correct answer from the options given below:
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(DROPPED)
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The electrostatic potential on the surface of uniformly charged spherical shell of radius $R = 10 \ cm$ is $120 \ V$. The potential at the centre of shell, at a distance $r = 5 \ cm$ from centre, and at a distance $r = 15 \ cm$ from the centre of the shell respectively, are:
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(Given $\mu_0 = 4\pi \times 10^{-7}$ Tm $A^{-1}$)
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Important Questions from Electricity and Magnetism
Two blocks of masses $m$ and $M$, ($M > m$), are placed on a frictionless table as shown in figure. A massless spring with spring constant $k$ is attached with the lower block. If the system is slightly displaced and released, then
A. The time period of small oscillation of the two blocks is $T = 2\pi \sqrt{\frac{(m+M)}{k}}$
B. The acceleration of the blocks is $a = \frac{kx}{M+m}$ (x = displacement of the blocks from the mean position)
C. The magnitude of the frictional force on the upper block is $\frac{\mu m|x|}{M+m}$
D. The maximum amplitude of the upper block, if it does not slip, is $\frac{\mu (M+m)g}{k}$
E. Maximum frictional force can be $\mu(M + m) g$.
Choose the correct answer from the options given below:
A wire of length $25 \ m$ and cross-sectional area $5 \ mm^2$ having resistivity of $2 \times 10^{-6} \ \Omega \ m$ is bent into a complete circle. The resistance between diametrically opposite points will be
(DROPPED)
A parallel plate capacitor is filled equally(half) with two dielectrics of dielectric constants $\varepsilon_1$ and $\varepsilon_2$, as shown in figures. The distance between the plates is $d$ and area of each plate is $A$. If capacitance in first configuration and second configuration are $C_1$ and $C_2$ respectively, then $\frac{C_1}{C_2}$ is :
First Configuration
The electrostatic potential on the surface of uniformly charged spherical shell of radius $R = 10 \ cm$ is $120 \ V$. The potential at the centre of shell, at a distance $r = 5 \ cm$ from centre, and at a distance $r = 15 \ cm$ from the centre of the shell respectively, are:
The radiation pressure exerted by a $450 \ W$ light source on a perfectly reflecting surface placed at $2m$ away from it, is
