Two small spherical balls of mass 10 g each with charges $-2 \mu \mathrm{C}$ and $2 \mu \mathrm{C}$, are attached to two ends of very light rigid rod of length 20 cm. The arrangement is now placed near an infinite nonconducting charge sheet with uniform charge density of $100 \mu \mathrm{C} / \mathrm{m}^{2}$ such that length of rod makes an angle of $30^{\circ}$ with electric field generated by charge sheet. Net torque acting on the rod is:
Two small spherical balls of mass 10 g each with charges $-2 \mu \mathrm{C}$ and $2 \mu \mathrm{C}$, are attached to two ends of very light rigid rod of length 20 cm. The arrangement is now placed near an infinite nonconducting charge sheet with uniform charge density of $100 \mu \mathrm{C} / \mathrm{m}^{2}$ such that length of rod makes an angle of $30^{\circ}$ with electric field generated by charge sheet. Net torque acting on the rod is:
Show Hint
- 112 Nm
- 1.12 Nm
- 2.24 Nm
- 11.2 Nm
The Correct Option is B
Approach Solution - 1
2. Torque acting on the rod: \[ \tau = PE \sin \theta \] \[ \tau = \left[ 2 \times 10^{-6} \times \frac{2}{10} \right] \left[ \frac{100 \times 10^{-6}}{2 \times 8.85 \times 10^{-12}} \right] \frac{1}{2} \] \[ \tau = \frac{10}{8.85} = 1.12 \mathrm{~Nm} \] Therefore, the correct answer is (2) 1.12 Nm.
Approach Solution -2
We are given two small spherical balls of equal mass \( m = 10 \, \mathrm{g} \) carrying charges \( q_1 = -2 \, \mu\mathrm{C} \) and \( q_2 = +2 \, \mu\mathrm{C} \). These are attached to the ends of a light rigid rod of length \( L = 20 \, \mathrm{cm} = 0.2 \, \mathrm{m} \). The arrangement is placed near an infinite non-conducting sheet having a uniform surface charge density \( \sigma = 100 \, \mu\mathrm{C/m^2} \). The rod makes an angle \( \theta = 30^\circ \) with the electric field generated by the sheet. We need to find the net torque acting on the rod.
Concept Used:
The electric field due to an infinite non-conducting sheet with surface charge density \( \sigma \) is:
\[ E = \frac{\sigma}{2\varepsilon_0} \]
The torque experienced by an electric dipole in a uniform electric field is:
\[ \tau = pE \sin\theta \]
where \( p = qL \) is the electric dipole moment.
Step-by-Step Solution:
Step 1: Compute the electric field \( E \) due to the sheet.
\[ E = \frac{\sigma}{2\varepsilon_0} \] \[ E = \frac{100 \times 10^{-6}}{2 \times 8.854 \times 10^{-12}} = 5.65 \times 10^{6} \, \mathrm{N/C} \]
Step 2: Calculate the dipole moment \( p \).
\[ p = qL = (2 \times 10^{-6})(0.2) = 4 \times 10^{-7} \, \mathrm{C \cdot m} \]
Step 3: Substitute values in torque formula.
\[ \tau = pE \sin\theta \] \[ \tau = (4 \times 10^{-7})(5.65 \times 10^{6}) \sin 30^\circ \]
Step 4: Simplify the expression.
\[ \tau = (2.26) \times 0.5 = 1.13 \, \mathrm{N \cdot m} \]
After rounding to two decimal places, the torque is approximately:
\[ \tau = 1.12 \, \mathrm{N \cdot m} \]
Final Computation & Result:
The net torque acting on the rod is:
\[ \boxed{\tau = 1.12 \, \mathrm{N \cdot m}} \]
Final Answer: \( 1.12 \, \mathrm{N \cdot m} \)
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Electrostatics Questions
- The resistivity (\( \rho \)) of a semiconductor varies with temperature. Which of the following curves represents the correct behavior?
- Given below are two statements : one is labelled a Assertion (A) and the other is labelled as Reason(R)
Assertion (A) : Work done by electric field on moving a positive charge on an equipotential surface is always zero.
Reason (R) : Electric lines of forces are always perpendicular to equipotential surfaces.
In the light of the above statements, choose the most appropriate answer from the options given below : - Two charges of \( -4 \, \mu C \) and \( +4 \, \mu C \) are placed at the points \( A(1, 0, 4) \, m \) and \( B(2, -1, 5) \, m \) located in an electric field \( \vec{E} = 0.20 \, \hat{i} \, V / cm \). The magnitude of the torque acting on the dipole is \( 8\sqrt{\alpha} \times 10^{-5} \, Nm \), where \( \alpha = \, \).
- An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet \( S \) having surface charge density \( +\sigma \). The electron at \( t = 0 \) is at a distance of 1 m from \( S \) and has a speed of 1 m/s. The maximum value of \( \sigma \) if the electron strikes \( S \) at \( t = 1 \, \text{s} \) is \( \alpha \left[ \frac{m \epsilon_0}{e} \right] \frac{\text{C}}{\text{m}^2} \). The value of \( \alpha \) is ______.
- Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of 37° with each other. When suspended in a liquid of density 0.7 g/cm3, the angle remains the same. If the density of the material of the sphere is 1.4 g/cm3, the dielectric constant of the liquid is ___________. (tan 37° =\( \frac{3}{4}\))
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.