Problem 27 - Master Resource Book in JEE Main Physics

Question

An electron of mass \(M_{e}\), initially at rest, moves through a certain distance in a uniform electric field in time \(t_{1}\). A proton of mass \(M_{p}\) also initially at rest, takes time \(t_{2}\) to move through an equal distance in this uniform electric field. Neglecting the effect of gravity, the ratio \(t_{2} / t_{1}\) is nearly equal to (a) 1 (b) \(\sqrt{M_{p} / M_{e}}\) (c) \(\sqrt{M_{e} / M_{p}}\) (d) 1836

Step-by-Step Solution

Answer

The ratio \( t_2 / t_1 \) is \( \sqrt{M_p / M_e} \), corresponding to option (b).

Step 1: Determine Electron Acceleration

The force on an electron due to the electric field is given by \( F = eE \), where \( e \) is the charge of the electron and \( E \) is the electric field strength. The acceleration \( a_e \) of the electron is then \( a_e = \frac{F}{M_e} = \frac{eE}{M_e} \).

Step 2: Determine Proton Acceleration

Similarly, for a proton, the force is \( F = eE \). Therefore, the acceleration of the proton \( a_p \) is \( a_p = \frac{F}{M_p} = \frac{eE}{M_p} \).

Step 3: Use Kinematic Equation for Time

Using the kinematic equation for an object starting from rest, the distance \( d \) moved by a particle with acceleration \( a \) in time \( t \) is \( d = \frac{1}{2} a t^2 \). Solving for time \( t \), we get \( t = \sqrt{\frac{2d}{a}} \).

Step 4: Express Time for Electron and Proton

Substitute the expressions for acceleration to find the time for electron \( t_1 = \sqrt{\frac{2d}{a_e}} = \sqrt{\frac{2dM_e}{eE}} \) and for the proton \( t_2 = \sqrt{\frac{2d}{a_p}} = \sqrt{\frac{2dM_p}{eE}} \).

Step 5: Determine the Ratio \( \frac{t_2}{t_1} \)

Calculate the ratio \( \frac{t_2}{t_1} \):\[\frac{t_2}{t_1} = \frac{\sqrt{\frac{2dM_p}{eE}}}{\sqrt{\frac{2dM_e}{eE}}} = \sqrt{\frac{M_p}{M_e}}\]Thus, the ratio \( \frac{t_2}{t_1} \) is equal to \( \sqrt{\frac{M_p}{M_e}} \).

Step 6: Choose the Correct Option

From the above calculation, the ratio \( \frac{t_2}{t_1} = \sqrt{\frac{M_p}{M_e}} \). This corresponds to option (b).

Key Concepts

Kinematics

Kinematics is the study of the motion of particles without focusing on the forces causing them. In this context, kinematics can help us understand how particles like electrons and protons move in an electric field. The motion can be described with concepts like velocity, acceleration, and displacement, neglecting the friction or other forces.

When a charged particle is at rest in an electric field, it starts to move due to the force exerted by the field. This motion can be described by kinematic equations. One important equation is the basic kinematic equation:

• The distance moved by a particle is given by: \[d = \frac{1}{2} a t^2\] where \(d\) is distance, \(a\) is acceleration, and \(t\) is the time taken.

The above equation is crucial to solving problems involving motion in an electric field when the initial velocity is zero, as is the case in many physics problems. Kinematics helps link the particle's acceleration to the time it travels through a certain distance.

Understanding kinematics empowers us to solve complex problems by breaking them down into simpler concepts such as time and distance, using their interrelation to find unknowns.

Electric Field

An electric field is a field around a charged particle within which other charged particles experience a force. It is a vector field and is denoted by \(E\).

Electric fields are responsible for exerting force on charged particles, such as electrons and protons, influencing their motion. This force can be calculated using the formula:

• Force \( F \) on a charge \( q \) in an electric field \( E \) is given by: \[F = qE\] where \(F\) is force, \(q\) is the charge of the particle, and \(E\) is the electric field strength.

This force initiates the movement of particles at rest in the field. For instance, both an electron and a proton experience this force due to their respective charges, although their masses differ significantly.

The electric field impacts the acceleration a particle experiences. This becomes crucial when comparing the time taken by different particles to move through the same distance, given the same field conditions. The particle's mass and charge are both essential in determining the behaviors within an electric field.

Particle Acceleration

Particle acceleration refers to the rate of change of velocity of a particle. This is important when analyzing particles in an electric field because it determines how quickly they move through space.

When a charged particle is subjected to an electric field, it accelerates in the direction of the field if positively charged, and in the opposite direction if negatively charged. The acceleration \(a\) can be calculated from the force exerted by the electric field:

• The acceleration \(a\) can be expressed as: \[a = \frac{F}{m} = \frac{qE}{m}\] where \(F\) is force, \(q\) is the charge, \(E\) is the electric field strength, and \(m\) is the mass of the particle.

This acceleration is different for electrons and protons due to their mass difference, even if they possess the same charge magnitude.

Particle acceleration plays a key role in their dynamics in a field, explaining how quickly they achieve a certain velocity and subsequently, how much time they take to cover known distances. Understanding acceleration helps in determining related kinematics aspects like velocity and displacement over time.