JEE Advanced 1982
LEVELJEE Advanced
Visualized Solution
The Sigma Insight: Motion of a Charge in Magnetic Fields
The Crossed-Field Setup
Imagine a charged particle entering a region where both an electric field and a magnetic field are present. This is a classic "crossed-field" setup, often used in velocity selectors and mass spectrometers. Our particle, with a specific mass and charge, enters this region at the origin, moving purely along the positive x-axis with a velocity m/s.
The electric field points downwards along the negative z-axis, while the magnetic field points along the positive y-axis. To understand the particle's journey, we must analyze the forces acting on it.
The Balancing Act
Lorentz Force
The total force on a charged particle in electromagnetic fields is given by the Lorentz force equation:
Let's break this down. First, the electric force . Substituting the given values, we find that the electric force pulls the particle downwards along the negative z-axis with a magnitude of N.
Now, what about the magnetic force? Using the right-hand rule for the cross product (where is along and is along ), we find that the magnetic force points upwards along the positive z-axis. Calculating its magnitude , we discover something remarkable: it is exactly N!
The electric and magnetic forces are perfectly balanced. They are equal in magnitude but opposite in direction, resulting in a net force of zero.
The Journey to Point P
Because the net force is zero, the particle experiences no acceleration. It continues to move in a straight line along the x-axis with its initial constant velocity.
We are asked to find its position at s. Using the simple kinematic equation for constant velocity, , we find:
At this moment, the particle is at point .
A Sudden Change
Circular Motion Begins
At exactly s, the electric field is abruptly switched off. The delicate balance is broken! The particle is now solely under the influence of the magnetic field.
Since the particle's velocity is perpendicular to the uniform magnetic field, it will immediately begin to move in a circular path. Because the velocity is along the x-axis and the magnetic field is along the y-axis, the magnetic force (and thus the circular path) will lie entirely in the x-z plane.
Let's determine the parameters of this circular motion. The radius is given by:
The angular velocity is:
Tracing the Semi-Circle
The particle travels in this circular path for the remaining time interval, .
How much of the circle does it trace in this time? We calculate the angle rotated, :
Notice that is approximately equal to radians (). This means the particle completes exactly a semi-circle!
Starting from point , it swings through the x-z plane. After a full semi-circle, its x-coordinate returns to m, its y-coordinate remains , and its z-coordinate increases by the diameter of the circle ( m).
Therefore, the final position of the particle is .
Similar Questions
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A particle of mass kg and charge C enters at in a region of uniform magnetic field of strength T along the direction shown in figure. The speed of the particle is m/s. (a) The magnetic field is directed along the inward normal to the plane of the paper. The particle leaves the region of the field at the point . Find the distance and the angle . (b) If the direction of the field is along the outward normal to the plane of the paper, find the time spent by the particle in the region of the magnetic field after entering it at .
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The region between and is filled with uniform steady magnetic field . A particle of mass , positive charge and velocity travels along -axis and enters the region of the magnetic field. Neglect the gravity throughout the question. (a) Find the value of if the particle emerges from the region of magnetic field with its final velocity at an angle to its initial velocity. (b) Find the final velocity of the particle and the time spent by it in the magnetic field, if the magnetic field now expands upto .
JEE Main 2003
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A particle of charge C moving with velocity along the X-axis enters a region where a magnetic field of induction is along the Y-axis and an electric field of magnitude is along the negative Z-axis. If the charged particle continues moving along the X-axis, the magnitude of is
(A)
(B)
(C)
(D)
JEE Advanced 2026
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In a vacuum chamber, a particle of charge and mass is projected with a velocity from the plane at time in an electric field of . At , the electric field is switched off and a magnetic field of is switched on. The acceleration due to gravity is . Correct option(s) is/are:
* Multiple Correct Options
(A)
The vertical distance of the particle from the plane at is .
(B)
The vertical distance of the particle from the plane at is .
(C)
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(D)
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JEE Main 2019
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An electron moving along the X-axis with an initial energy of , enters a region of magnetic field at (see figure). The field extends between and . The electron is detected at the point on a screen placed away from the point . The distance between and (on the screen) is (Take, electron's charge , mass of electron )
(A)
11.65 cm
(B)
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(C)
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(D)
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JEE Main 2019
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The region between and contains a magnetic field . A particle of mass and charge enters the region with a velocity . If , then the acceleration of the charged particle at the point of its emergence at the other side is
(A)
(B)
(C)
(D)
JEE Advanced 2013
LEVELJEE Advanced
A particle of mass and positive charge , moving with a constant velocity , enters a region of uniform static magnetic field normal to the - plane. The region of the magnetic field extends from to for all values of . After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity . The correct statement(s) is (are)
* Multiple Correct Options
(A)
the direction of the magnetic field is direction.
(B)
the direction of the magnetic field is direction
(C)
the magnitude of the magnetic field is units.
(D)
the magnitude of the magnetic field is units.
JEE Main 2020
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An electron is moving along +x-direction with a velocity of . It enters a region of uniform electric field of pointing along +y-direction. The magnitude and direction of the magnetic field set up in this region such that the electron keeps moving along the x-direction will be
(A)
, along + z-direction
(B)
, along − z-direction
(C)
, along + z-direction
(D)
, along − z-direction
JEE Advanced 2017
LEVELJEE Advanced
Comprehension Passage
A charged particle (electron or proton) is introduced at the origin () with a given initial velocity . A uniform electric field and a uniform magnetic field exist everywhere. The velocity , electric field and magnetic field are given in columns 1, 2 and 3, respectively. The quantities are positive in magnitude.
$\begin{array}{lll}
\hline \text{Column 1} & \text{Column 2} & \text{Column 3} \\
\hline \text{(I) Electron with } \mathbf{v} = 2\frac{E_0}{B_0}\hat{x} & \text{(i) } \mathbf{E} = E_0\hat{z} & \text{(P) } \mathbf{B} = -B_0\hat{x} \\
\text{(II) Electron with } \mathbf{v} = \frac{E_0}{B_0}\hat{y} & \text{(ii) } \mathbf{E} = -E_0\hat{y} & \text{(Q) } \mathbf{B} = B_0\hat{x} \\
\text{(III) Proton with } \mathbf{v} = 0 & \text{(iii) } \mathbf{E} = -E_0\hat{x} & \text{(R) } \mathbf{B} = B_0\hat{y} \\
\text{(IV) Proton with } \mathbf{v} = 2\frac{E_0}{B_0}\hat{x} & \text{(iv) } \mathbf{E} = E_0\hat{x} & \text{(S) } \mathbf{B} = B_0\hat{z} \\
\hline
\end{array}$
Question 1:
In which case would the particle move in a straight line along the negative direction of Y-axis (i.e. move along )?
(A)
(IV) (ii) (S)
(B)
(II) (iii) (Q)
(C)
(III) (ii) (R)
(D)
(III) (ii) (P)
Question 2:
In which case will the particle move in a straight line with constant velocity?
(A)
(II) (iii) (S)
(B)
(III) (iii) (P)
(C)
(IV) (i) (S)
(D)
(III) (ii) (R)
Question 3:
In which case will the particle describe a helical path with axis along the positive z-direction?
(A)
(II) (ii) (R)
(B)
(III) (iii) (P)
(C)
(IV) (i) (S)
(D)
(IV) (ii) (R)