Problem 13 - Master Resource Book in JEE Main Physics

Question

A charged dust particle of radius \(5 \times 10^{-7} \mathrm{~m}\) is located in a horizontal electric field having an intensity of \(6.28 \times 10^{5} \mathrm{Vm}^{-1}\). The surrounding medium in air with coefficient of viscosity \(\eta=16 \times 10^{-15} \mathrm{Nsm}^{-2}\). If this particle moves with a uniform horizontal speed of \(0.01 \mathrm{~ms}^{-1}\), the number of electrons on it will be (a) 20 (b) 15 (c) 25 (d) 30

Step-by-Step Solution

Answer

The number of electrons on the particle is 25 (option c).

Step 1: Identify Forces on the Particle

The charged dust particle is in equilibrium, moving at a uniform speed. Therefore, the electric force (F_e) acting on it is balanced by the viscous drag force (F_d). We can express this balance as:\[ F_e = F_d \]

Step 2: Express Electric Force

The electric force F_e on the charged particle is given by the product of its charge (q) and the electric field intensity (E):\[ F_e = qE \]

Step 3: Express Viscous Drag Force

According to Stokes' Law, the viscous drag force F_d is determined by the formula:\[ F_d = 6 \pi \eta rv \] where \eta is the viscosity of air, r is the radius of the particle, and v is the particle's speed.

Step 4: Set Forces Equal and Solve for Charge

Since F_e = F_d, we can equate the formulas for electric force and viscous drag force:\[ qE = 6 \pi \eta rv \]Solving for q gives:\[ q = \frac{6 \pi \eta rv}{E} \]

Step 5: Calculate the Charge q

Substitute the given values into the equation:\[ q = \frac{6 \pi (16 \times 10^{-15}) (5 \times 10^{-7}) (0.01)}{6.28 \times 10^{5}} \]Simplify to find q.

Step 6: Convert Charge to Number of Electrons

Use the formula q = ne to find the number of electrons (n), where e is the elementary charge (e = 1.6 \times 10^{-19} C):\[ n = \frac{q}{e} \]Substitute the calculated q and solve for n.

Key Concepts

Electric Fields

An electric field is a region around a charged object where other charges experience a force. It's an invisible force field around charged particles. The strength of this field is defined by its intensity, often represented as \( E \), which tells us how strong the field is in a particular direction.When a charged particle enters an electric field, it experiences a force due to this field. The force can be attractive or repulsive, depending on the type of charge of the particle (positive or negative). The relation for calculating the electric force \( F_e \) is given by \( F_e = qE \), where \( q \) is the charge of the particle.So, when you have a particle in a horizontal electric field, like in our exercise, it's experiencing a horizontal force due to the electric field. This is part of what's causing it to move.

Viscosity and Drag Forces

Viscosity is a measure of a fluid's resistance to deformation. It's like the 'thickness' or 'stickiness' of a fluid. In our scenario, the air acts as the fluid surrounding the dust particle. This air exerts a drag force on the moving particle, which is opposed to its motion.The viscous drag force, often denoted as \( F_d \), can be calculated using Stokes' Law for small spherical particles moving through a viscous medium. The formula is \( F_d = 6\pi\eta rv \), where:

• \( \eta \) is the fluid's viscosity.
• \( r \) is the radius of the particle.
• \( v \) is the velocity of the particle through the fluid.

This drag force is an important factor because it balances out the electric force, allowing the particle to move with a uniform speed.

Charge of Particles

Charge is a fundamental property of matter that causes it to experience a force in an electric field or magnetic field. It's measured in coulombs, and it can either be positive or negative, depending on the presence of protons or electrons.To find out how many electrons are present on the particle in our exercise, we first need to figure out the total charge \( q \) on the particle. Once let's call this calculated charge \( q \). Then, to find how many electrons contribute to this charge, we use the relation \( n = \frac{q}{e} \), where \( e \) is the elementary charge with a known value of \( 1.6 \times 10^{-19} \text{ C} \).Why electrons? Because each electron carries a negative charge, and multiple electrons will add up to the total negative charge observed on the particle.

Stokes' Law

Stokes' Law is a scientific principle used to determine the drag force experienced by a spherical object moving through a viscous fluid. This is quite useful for small particles like the dust particle in our exercise.According to Stokes' Law, when a sphere moves through a viscous medium, it experiences a retardation force due to viscosity. The law is mathematically represented as: \[ F_d = 6 \pi \eta rv \]This considers:

• \( \eta \) - the viscosity of the medium (how thick the air is).
• \( r \) - the radius of the sphere, indicating how small or large it is.
• \( v \) - the velocity, or how fast it’s moving through the fluid.

Stokes' Law helps us understand how drag and resistance are calculated, enabling us to solve for the charge using given values.