Problem 150 - Master Resource Book in JEE Main Physics

Question

Two small spheres of masses, \(M_{1}\) and \(M_{2}\) are suspended by weightless insulating threads of lengths \(L_{1}\) and \(L_{2} .\) The sphere carry charges \(Q_{1}\) and \(Q_{2}\), respectively. The spheres are suspended such that they are in level with another and the threads are inclined to the vertical at angles of \(\theta_{1}\) and \(\theta_{2}\) as shown below, which one of the following conditions is essential, if \(Q_{1}=Q_{2}\) ? (a) \(M_{1} \neq M_{2}\) but \(Q_{1}=Q_{2}\) (b) \(M_{1}=M_{2}\) (c) \(Q_{1}=Q_{2}\) (d) \(L_{1}=L_{2}\)

Step-by-Step Solution

Answer

Condition (a) is essential: \(M_{1} \neq M_{2}\), but \(Q_{1} = Q_{2}\).

Step 1: Understanding the forces in equilibrium

In this scenario, the spheres are in equilibrium due to the forces acting on them. Each sphere has gravitational force downward, a tension force upward along the direction of the thread, and an electrostatic force between the two spheres. Since the spheres are charged and repel each other, the horizontal component of the tension must balance the electrostatic force.

Step 2: Analyzing the electrostatic force

The electrostatic force acting on each sphere due to their charges is given by Coulomb's Law: \[ F = k \frac{Q_1 Q_2}{r^2} \]where \(k\) is Coulomb's constant, \(Q_1\) and \(Q_2\) are the charges, and \(r\) is the distance between the charges. Since \(Q_1 = Q_2\), the electrostatic force depends equally on the charges of the two spheres.

Step 3: Examining the tension and its components

The tension in the thread can be decomposed into two components: vertical and horizontal. The vertical component balances the weight of the sphere, and the horizontal component balances the electrostatic force. For sphere 1, the tension \(T_1\) is resolved into:\[ T_1 \cos(\theta_1) = M_1 g \] (balances weight)\[ T_1 \sin(\theta_1) = F \] (balances electrostatic force)The same analysis applies to sphere 2.

Step 4: Analyzing equilibrium and determining conditions

In equilibrium, the horizontal forces \(T_1 \sin(\theta_1)\) and \(T_2 \sin(\theta_2)\), which are due to the electrostatic force, must be equal for both spheres because the charge magnitude is the same: \[ M_1 g \tan(\theta_1) = M_2 g \tan(\theta_2) \] This shows that the mass ratio and angle ratio must satisfy the equality condition. Since we are examining under the condition \(Q_1 = Q_2\), the only way to maintain equilibrium is if the threads have equal angles, meaning \(M_1 eq M_2\) but equal charges, given \(Q_1 = Q_2\).

Key Concepts

Coulomb's Law

Coulomb's Law is a foundation in electrostatics, describing how charged objects interact with one another. It's all about the force between two charged spheres. The force depends on the product of the charges and the distance between the charges. It's given by the formula:

• \[ F = k \frac{Q_1 Q_2}{r^2} \]

Here, \(k\) is Coulomb's constant, \(Q_1\) and \(Q_2\) are the charges, and \(r\) is the distance between the charges.

When the charges are equal, like in this scenario, the force is proportional to the inverse square of the distance between them. This law helps us understand the push or pull that kept these spheres apart due to similarly charged bodies repelling each other.

Tension in Strings

Tension in strings plays a crucial role when analyzing objects suspended in equilibrium. When a sphere is hanging on a thread, tension goes to work, countering other forces acting on the sphere. It has two main components:

• Vertical Component: It balances the weight of the sphere.
• Horizontal Component: It counters the electrostatic force.

The tension \(T\) is resolved into these components:

• Vertical: \(T \cos(\theta) = Mg\)
• Horizontal: \(T \sin(\theta) = F\)

where \(M\) is the mass of the sphere, \(g\) is the acceleration due to gravity, \(\theta\) is the angle of the string with the vertical, and \(F\) is the electrostatic force. Understand these components aids in figuring out how the thread supports the spheres against both gravitational pull and inter-sphere repulsion.

Forces in Equilibrium

The idea of forces in equilibrium is that all the forces acting on an object balance each other out. This creates a steady state where there is no net force, and hence no change in motion. In the case with the spheres, several forces are at play:

• Gravitational Force: Acting downwards, due to the mass of the spheres.
• Tension: In the string, divided into vertical and horizontal components.
• Electrostatic Force: Acting horizontally due to the charges on the spheres.

For equilibrium, the condition is:

• Vertical equilibrium: \(T \cos(\theta) = Mg\)
• Horizontal equilibrium: \(T \sin(\theta) = F\)
• Overall equilibrium condition: \(M_1 g \tan(\theta_1) = M_2 g \tan(\theta_2)\)

Ensuring these holds true signifies the forces are balanced, thus forming an equilibrium state.

Mass and Charge Ratios

Understanding the mass and charge ratios in such problems helps in ensuring the equilibrium of forces acting on charged bodies. When the charges \(Q_1\) and \(Q_2\) are equal, we dive into the mass and angle aspects for maintaining equilibrium.We've shown earlier that the equilibrium condition reflects that:

• \(M_1 g \tan(\theta_1) = M_2 g \tan(\theta_2)\)

since \(Q_1 = Q_2\). This implies the ratios of the tangent of the angles must equal the ratios of the masses.

• If the mass \(M_1\) is not equal to \(M_2\), the angles \(\theta_1\) and \(\theta_2\) should adjust to balance.

Understanding these relationships helps in deducing the conditions under which suspended charges would remain in static harmony, even when their masses differ.