Advertisements
Advertisements
प्रश्न
Two particles, each with mass m are placed at a separation d in a uniform magnetic field B, as shown in the figure. They have opposite charges of equal magnitude q. At time t = 0, the particles are projected towards each other, each with a speed v. Suppose the Coulomb force between the charges is switched off. (a) Find the maximum value vmof the projection speed, so that the two particles do not collide. (b) What would be the minimum and maximum separation between the particles if v = vm/2? (c) At what instant will a collision occur between the particles if v = 2vm? (d) Suppose v = 2vm and the collision between the particles is completely inelastic. Describe the motion after the collision.
Advertisements
उत्तर
Given,
Mass of two particles = m
Distance between them = d
Both the particles have equal charges in magnitude but opposite polarity equal to q.
As per the question, both the particles are projected towards each other with equal speed v.
It is assumed that Coulomb force between the charges is switched off.
(a) The maximum value vm of the projection speed so that the two particles do not collide:-
Both the particles will not collide if
d = r1 + r2 (where, r1 = r2 = radius of circular orbit described by the charged particles)
`⇒ d = (mv_m)/(qB) + (mv_m)/(qB) = (2mv_m)/(qB)`
`so, v_m = (qBd)/(2m)`
(b) The minimum and maximum separation between the particles if v = vm/2:-
Let the radius of the curved path taken by the particles, when they are projected with speed vm/2, be r.
So, minimum separation between the particles = (d - 2r)
`⇒ (d - 2r) = (2mv_m)/(qB) - (2mv)/(qB)`
`⇒ (d - 2r) = (mv_m)/(qB)`
`⇒ (d - 2r) = d/2`
Maximum distance separation = (d + 2r)
`⇒ d + 2r = d + d/2 = (3d)/2`
(c) The instant at which the collision occurs between the particles when
The particles will collide at a distance d/2 along the horizontal direction.
Let they collide after time t.
Velocity of the particles along the horizontal direction will remain the same.
Therefore, `t = d//2/(2v_m)`
`⇒ t = d/4xx(2m)/(qBd)`
(d) The motion of the two particles after collision when the collision is completely inelastic:-
v = 2vm
Let the particles collide at point P.
And at point P, both the particles will have motion in upward direction.
As the collision is inelastic they stick together.
Distance between centres = d
Velocity along the horizontal direction does not get affected due to the magnetic force.
At point P, velocities along the horizontal direction are equal and opposite. So, they cancel each other.
Velocity along the vertical direction (upward) will add up.
Magnetic force acting along the vertical direction,
`F=q(2v_m)B`
Acceleration along the vertical direction,
`a = F/m = (2qv_mB)/m`
Velocity of the combined mass at point P is along the vertical direction. So,
`v'= 0 + a × t`
`v' = 0 + ((2qv_mB)/(m))xx((m)/(2qb))`
`v' = v_m`
Hence, both the particles will behave as a combined mass and move with velocity vm.
APPEARS IN
संबंधित प्रश्न
Show that the kinetic energy of the particle moving in a magnetic field remains constant.
An electron moving horizontally with a velocity of 4 ✕ 104 m/s enters a region of uniform magnetic field of 10−5 T acting vertically upward as shown in the figure. Draw its trajectory and find out the time it takes to come out of the region of magnetic
field.
A flexible wire of irregular shape, abcd, as shown in the figure, turns into a circular shape when placed in a region of magnetic field which is directed normal to the plane of the loop away from the reader. Predict the direction of the induced current in the wire.
Assume that the magnetic field is uniform in a cubical region and zero outside. Can you project a charged particle from outside into the field, so that the particle describes a complete circle in the field?
A positively-charged particle projected towards east is deflected towards north by a magnetic field. The field may be
A charged particle is whirled in a horizontal circle on a frictionless table by attaching it to a string fixed at one point. If a magnetic field is switched on in the vertical direction, the tension in the string
An electric current i enters and leaves a uniform circular wire of radius a through diametrically opposite points. A charged particle q, moving along the axis of the circular wire, passes through its centre at speed v. The magnetic force acting on the particle, when it passes through the centre, has a magnitude equal to
If a charged particle at rest experiences no electromagnetic force,
(a) the electric field must be zero
(b) the magnetic field must be zero
(c) the electric field may or may not be zero
(d) the magnetic field may or may not be zero
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.
Using the formula \[\vec{F} = q \vec{v} \times \vec{B} \text{ and } B = \frac{\mu_0 i}{2\pi r}\]show that the SI units of the magnetic field B and the permeability constant µ0 may be written as N mA−1 and NA−2 respectively.
A semicircular wire of radius 5.0 cm carries a current of 5.0 A. A magnetic field B of magnitude 0.50 T exists along the perpendicular to the plane of the wire. Find the magnitude of the magnetic force acting on the wire.
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 current i is passed through a silver strip of width d and area of cross-section A. The number of free electrons per unit volume is n. (a) Find the drift velocity v of the electrons. (b) If a magnetic field B exists in the region, as shown in the figure, what is the average magnetic force on the free electrons? (c) Due to the magnetic force, the free electrons get accumulated on one side of the conductor along its length. This produces a transverse electric field in the conductor, which opposes the magnetic force on the electrons. Find the magnitude of the electric field which will stop further accumulation of electrons. (d) What will be the potential difference developed across the width of the conductor due to the electron-accumulation? The appearance of a transverse emf, when a current-carrying wire is placed in a magnetic field, is called Hall effect.
A circular coil of radius 2.0 cm has 500 turns and carries a current of 1.0 A. Its axis makes an angle of 30° with the uniform magnetic field of magnitude 0.40 T that exists in the space. Find the torque acting on the coil.
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 particle of mass m and charge q is projected into a region that has a perpendicular magnetic field B. Find the angle of deviation (figure) of the particle as it comes out of the magnetic field if the width d of the region is very slightly smaller than
(a) `(mv)/(qB)` (b)`(mv)/(2qB)` (c)`(2mv)/(qB)`
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.
