Problem 18 - Master Resource Book in JEE Main Physics

Question

An electric field of intensity \(6 \times 10^{4} \mathrm{Vm}^{-1}\) is applied perpendicular to the direction of motion of the electron. A magnetic field of induction \(8 \times 10^{-2} \mathrm{Wm}^{-2}\) is applied perpendicular to both the electric field and direction of motion of the electron. What is the velocity of the electron if it passes undeflected? (a) \(7.5 \times 10^{5} \mathrm{~ms}^{-1}\) (b) \(7.5 \times 10^{-5} \mathrm{~ms}^{-1}\) (c) \(48 \times 10^{-2} \mathrm{~ms}^{-1}\) (d) It is never possible

Step-by-Step Solution

Answer

(a) \(7.5 \times 10^5 \mathrm{~ms}^{-1}\)

Step 1: Understanding the Conditions for Zero Deflection

An electron moving in electric and magnetic fields without deflection means that the magnetic force and the electric force are balanced and act in opposite directions. Thus, the magnitude of these forces must be equal.

Step 2: Expressing Forces from Fields

The force due to the electric field is given by \( F_e = eE \), where \( e \) is the charge of the electron, and \( E \) is the electric field intensity. Similarly, the force due to the magnetic field is \( F_m = evB \), where \( v \) is the speed of the electron, and \( B \) is the magnetic field induction.

Step 3: Setting Up the Equation for Undeflected Motion

Since the electron passes undeflected, the forces are equal: \[ eE = evB \]. We can simplify this equation to \[ E = vB \], since \( e \), the electron's charge, is present on both sides of the equation.

Step 4: Solving for Velocity

Rearrange the equation from the previous step to find the velocity \( v \): \[ v = \frac{E}{B} \]. Substitute \( E = 6 \times 10^4 \text{ Vm}^{-1} \) and \( B = 8 \times 10^{-2} \text{ Wb/m}^2 \) into the equation: \[ v = \frac{6 \times 10^4}{8 \times 10^{-2}} = 7.5 \times 10^5 \text{ ms}^{-1} \].

Step 5: Identifying the Correct Answer

The velocity calculated matches the option (a): \( 7.5 \times 10^5 \text{ ms}^{-1} \). Therefore, (a) is the correct answer.

Key Concepts

Electric Field

An electric field is a fundamental concept in physics that describes the force field surrounding electric charges. It is represented by the symbol **E**. In the context of electromagnetic induction, the electric field exerts a force on charged particles, such as electrons. This force is calculated using the formula:

• \( F_e = eE \),

where \( e \) is the charge of the electron and \( E \) is the electric field intensity.

The electric field intensity is measured in volts per meter (Vm\(^{-1}\)). It describes how strong the field is and in which direction it will push positive charges. In an electric field perpendicular to an electron's motion, the field tries to change the electron's path. However, when balanced by other forces, such as a magnetic field, the electron can pass through undeflected.

Magnetic Field

A magnetic field is another key player in the realm of electromagnetism. It exerts a force on moving charges, such as current-carrying wires or electrons in motion. The magnetic field is represented by the symbol **B** and is measured in Weber per square meter (Wb/m\(^{2}\)).

In electromagnetic induction, when an electron moves through a magnetic field perpendicular to its velocity, the field exerts a force given by the equation:

• \( F_m = evB \),

where \( v \) is the velocity of the electron. The direction of this magnetic force is given by the right-hand rule, perpendicular to both the magnetic field and the electron's velocity.

In scenarios where an electron experiences both an electric and a magnetic field, the interaction between these fields can result in the electron moving without deflection, provided the forces from each field cancel each other out.

Lorentz Force

The Lorentz force describes the total force exerted on a charged particle moving in electric and magnetic fields. It is a combination of the forces due to the electric and magnetic fields and is described by the formula:

• \( F = F_e + F_m = eE + evB \),

where \( F_e \) is the electric force and \( F_m \) is the magnetic force.

For an electron to travel undeflected through the combined fields, the Lorentz force must be zero. This condition is achieved when the forces from the electric and magnetic fields are equal in magnitude but opposite in direction:

• \( eE = evB \).

Solving this equation gives the velocity \( v \) at which the electron remains undeflected:

• \( v = \frac{E}{B} \).

This balance of forces makes it possible for the charged particle to move through a region of space influenced by both fields without deviation, illustrating a fundamental concept of electromagnetic induction.