Problem 74 - Arihant AIEEE Physics

Question

A copper ball of density \(\rho_{c}\) and diameter \(d\) is immersed in oil of density \(\rho_{\theta}\). What charge should be present on the ball, so that it could be suspended in the oil, if a homogeneous electric field \(E\) is applied vertically upward? (a) \(Q=\frac{\pi d^{2}\left(\rho_{c}-\rho_{0}\right) g}{6 E}\) (b) \(Q=\frac{\pi d^{\vec{j}}\left(\rho_{c}-\rho_{a}\right) g}{6 E}\) (c) \(Q=\frac{\pi d^{3}\left(\rho_{c}-\rho_{0}\right) g}{E}\) (d) None of these

Step-by-Step Solution

Answer

The correct answer is (a): \( Q=\frac{\pi d^3 (\rho_c-\rho_\theta) g}{6 E} \).

Step 1: Understanding the Forces

To find the charge needed to suspend the ball in equilibrium, we must balance the gravitational force, buoyant force, and the electric force. The weight of the copper ball is given by the gravitational force: \( F_g = V \cdot \rho_c \cdot g \), where \( V \) is the volume of the ball, \( \rho_c \) is the density of copper, and \( g \) is the acceleration due to gravity.

Step 2: Calculate the Volume of the Ball

The volume \( V \) of the sphere (copper ball) is calculated using the formula for the volume of a sphere: \( V = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 = \frac{\pi d^3}{6} \).

Step 3: Calculate the Buoyant Force

The buoyant force exerted by the oil is \( F_b = V \cdot \rho_\theta \cdot g \), where \( \rho_\theta \) is the density of the oil.

Step 4: Identify the Electric Force

The electric force acting on the charged ball in an electric field \( E \) is \( F_e = Q \cdot E \), where \( Q \) is the charge we want to determine.

Step 5: Setting up the Equilibrium Condition

For the ball to be suspended and not sink or rise, the sum of forces must be zero: \( F_g - F_b + F_e = 0 \). This gives us: \( \rho_c V g - \rho_\theta V g + Q E = 0 \).

Step 6: Solving for the Charge Q

Rearrange the equilibrium equation to solve for \( Q \): \( Q = (\rho_c - \rho_\theta) V g / E \). Substitute the volume \( V = \frac{\pi d^3}{6} \): \( Q = \frac{\pi d^3 (\rho_c - \rho_\theta) g}{6 E} \).

Step 7: Compare with Given Options

The expression derived \( Q = \frac{\pi d^3 (\rho_c - \rho_\theta) g}{6 E} \) matches option (a) when correctly interpreted since the subscripts are correct for the density difference between copper and oil.

Key Concepts

Electric Force

In simple terms, electric force is the force that occurs between two charged objects. It is a fundamental interaction that causes repulsion or attraction depending on the charges. For a charged object in an electric field, such as our copper ball, the electric force can be calculated using the formula:

• Electric Force, \( F_e = Q \cdot E \)

Here, \( Q \) is the charge on the object, and \( E \) is the electric field strength.
This force is crucial for achieving equilibrium in the problem because it works against gravitational and buoyant forces to hold the ball in place within the liquid.
The direction of this force is in the direction of the electric field if the charge is positive, and opposite for a negative charge.
Remember, the electric force must perfectly balance other forces acting on the object for equilibriums to be achieved.

Buoyant Force

The buoyant force is an upward force exerted by a fluid that opposes the weight of an object immersed in it. This force is described by Archimedes' principle, which states that the upward buoyant force is equal to the weight of the fluid displaced by the object.
In our scenario, the buoyant force \( F_b \) is given by:

• \( F_b = V \cdot \rho_\theta \cdot g \)

Here, \( V \) is the volume of the copper ball, \( \rho_\theta \) is the density of the oil, and \( g \) is the acceleration due to gravity.
The buoyant force is crucial in counteracting the gravitational force and partially supporting the object's weight, making it easier for the additional force, like the electric force, to achieve balance (equilibrium) in fluids.

Gravitational Force

Gravitational force is a natural phenomenon by which all things with mass or energy are brought toward one another, including our copper ball and Earth. This force can be described with Newton's law of universal gravitation.
For the copper ball, the gravitational force \( F_g \) is calculated as follows:

• \( F_g = V \cdot \rho_c \cdot g \)

Where \( V \) is the volume of the ball, \( \rho_c \) is the density of the copper, and \( g \) is acceleration due to gravity.
This downward force must be considered when balancing the forces to keep the ball suspended. In equilibrium, this force is balanced by the combined upward forces—the buoyant force and the electric force.

Density

Density is a measure of how much mass is contained in a given volume. It plays a significant role in understanding buoyancy and overall equilibrium in fluids. In the given scenario, we deal with the density of two different materials: copper of the ball and oil.

• Copper Density \( \rho_c \)
• Oil Density \( \rho_\theta \)

The difference in these densities \( (\rho_c - \rho_\theta) \) determines the net downward gravitational force after accounting for buoyancy.
Higher densities result in heavier materials, which creates a greater gravitational force requiring a corresponding increase in other balancing forces, such as electric force, to maintain equilibrium.

Equilibrium in Fluids

Equilibrium occurs when all forces acting on an object in a fluid are perfectly balanced, preventing the object from sinking or rising. This is a critical condition for our copper ball suspended in oil within an electric field.
To achieve equilibrium:

• The net force acting on the ball must be zero.
• The gravitational force pulling down should equal the combined buoyant and electric forces pushing up.

Formulaically, this balance is expressed as:

• \( F_g - F_b + F_e = 0 \)

Understanding equilibrium requires assessing these forces and ensuring their correct measurements and calculations meet this condition.
Thus, equilibrium is all about achieving a state where no net motion occurs, crucial for analysis in physics when dealing with systems or objects within fluids.