Advertisements
Advertisements
Question
Electrons emitted with negligible speed from an electron gun are accelerated through a potential difference V along the x-axis. These electrons emerge from a narrow hole into a uniform magnetic field B directed along this axis. However, some of the electrons emerging from the hole make slightly divergent angles, as shown in the figure. Show that these paraxial electrons are refocussed on the x-axis at a distance `sqrt(8pi^2mV)/(eB^2).`
Advertisements
Solution
Given:-
Electrons are accelerated through a potential difference = V
Let the mass of an electron be m and the charge of an electron be e.
We know:-
Electric field, E = `V/r`
Force experienced by the electron, F = eE
Acceleration of the electron, a = `(eV)/(rm)`
Using the equation of motion
v2 − u2= 2 × a × s,
v2 = 2 × a × s (As u = 0)
Here, s = r
v =`sqrt(2eVr)/(rm)`
= `sqrt(2eV)/m`
Time taken by electron to cover the curved path,
As the acceleration of the electron is along the y axis only, it travels along the x axis with uniform velocity.
Velocity of the electron moving along the field remains v.
Therefore, the distance at which the beam is refocused, d = v × T
`d = sqrt(2eV)/m`
`d = sqrt(8pi^2mV)/(eB^2)`
APPEARS IN
RELATED QUESTIONS
A long horizontal wire P carries a current of 50A. It is rigidly fixed. Another wire Q is placed directly above and parallel to P, as shown in Figure 1 below. The weight per unit length of the wire Q is 0.025 Nm-1 and it carries a current of 25A. Find the distance 'r' of the wire Q from the wire P so that the wire Q remains at rest
Write the expression for the force,`vecF` acting on a charged particle of charge ‘q’, moving with a velocity `vecV` in the presence of both electric field `vecF`and magnetic field `vecB` . Obtain the condition under which the particle moves undeflected through the fields.
Write the expression for Lorentz magnetic force on a particle of charge ‘q’ moving with velocity `vecv` in a magnetic field`vecB`. Show that no work is done by this force on the charged particle.
A positively-charged particle projected towards east is deflected towards north by a magnetic field. The field may be
A beam consisting of protons and electrons moving at the same speed goes through a thin region in which there is a magnetic field perpendicular to the beam. The protons and the electrons
If a charged particle moves unaccelerated in a region containing electric and magnetic fields
(a) `vecE "must be perpendicular" to vecB`
(b) `vecv "must be perpendicular" to vecE`
(c) must be perpendicular to v_B
A particle is projected in a plane perpendicular to a uniform magnetic field. The area bounded by the path described by the particle is proportional to
Two particles X and Y having equal charge, after being accelerated through the same potential difference enter a region of uniform magnetic field and describe circular paths of radii R1 and R2 respectively. The ratio of the mass of X to that of Y is ______.
A 10 g bullet with a charge of 4.00 μC is fired at a speed of 270 m s−1 in a horizontal direction. A vertical magnetic field of 500 µT exists in the space. Find the deflection of the bullet due to the magnetic field as it travels through 100 m. Make appropriate approximations.
Prove that the force acting on a current-carrying wire, joining two fixed points a and b in a uniform magnetic field, is independent of the shape of the wire.
A wire, carrying a current i, is kept in the x−y plane along the curve y = A sin `((2x)/lamda x)`. magnetic field B exists in the z direction. Find the magnitude of the magnetic force on the portion of the wire between x = 0 and x = λ.
A metal wire PQ of mass 10 g lies at rest on two horizontal metal rails separated by 4.90 cm (figure). A vertically-downward magnetic field of magnitude 0.800 T exists in the space. The resistance of the circuit is slowly decreased and it is found that when the resistance goes below 20.0 Ω, the wire PQ starts sliding on the rails. Find the coefficient of friction.
A charged particle is accelerated through a potential difference of 12 kV and acquires a speed of 1.0 × 106 m s−1. It is then injected perpendicularly into a magnetic field of strength 0.2 T. Find the radius of the circle described by it.
A narrow beam of singly-charged carbon ions, moving at a constant velocity of 6.0 × 104m s−1, is sent perpendicularly in a rectangular region of uniform magnetic field B = 0.5 T (figure). It is found that two beams emerge from the field in the backward direction, the separations from the incident beam being 3.0 cm and 3.5 cm. Identify the isotopes present in the ion beam. Take the mass of an ion = A(1.6 × 10−27) kg, where A is the mass number.
A narrow beam of singly charged potassium ions of kinetic energy 32 keV is injected into a region of width 1.00 cm with a magnetic field of strength 0.500 T, as shown in the figure. The ions are collected at a screen 95.5 cm away from the field region. If the beam contains isotopes of atomic weights 39 and 41, find the separation between the points where these isotopes strike the screen. Take the mass of a potassium ion = A (1.6 × 10−27) kg, where A is the mass number.
The figure shows a convex lens of focal length 12 cm lying in a uniform magnetic field Bof magnitude 1.2 T parallel to its principal axis. A particle with charge 2.0 × 10−3 C and mass 2.0 × 10−5 kg is projected perpendicular to the plane of the diagram with a speed of 4.8 m s−1. The particle moves along a circle with its centre on the principal axis at a distance of 18 cm from the lens. Show that the image of the particle moves along a circle and find the radius of that circle.
A long, straight wire carrying a current of 30 A is placed in an external, uniform magnetic field of 4.0 × 10−4 T parallel to the current. Find the magnitude of the resultant magnetic field at a point 2.0 cm away from the wire.
The velocity of a body of mass 2 kg as a function of time t is given by v(t) = 2t`hat"i" + "t"^2hat"j"`. The force acting on it, at time t = 2 s is given by ______.
A particle of mass 10 mg and having a charge of 50 mC is projected with a speed of 15 m/s into a uniform magnetic field of 125 mT. Assuming that the particle is projected with its velocity perpendicular to the magnetic field, the time after which the particle reaches its original position for the first time is ______.
