CDS (II) 2023 General Knowledge · Question 36 of 120

Correct — C, F is maximum when v and B are perpendicular to each other. The magnetic force on a moving charge is the vector product F = q(v × B), and its magnitude is F = qvB sin θ, where θ is the angle between the velocity and the magnetic field. Everything in that expression except sin θ is fixed once the charge, the speed and the field strength are fixed, so the whole angular behaviour of the force sits in the sine. The sine of an angle is largest at 90 degrees, where it equals one, giving the maximum possible force qvB. It is zero at 0 degrees and again at 180 degrees, so a charge moving exactly along the field, or exactly against it, feels no magnetic force at all and sails straight through undeflected. This is why a charged particle fired across a uniform field moves in a circle — the force stays perpendicular to the velocity, turning it without changing its speed — while one fired along the field is untouched, and one fired obliquely traces a helix, with the component of velocity along the field carrying it forward and the perpendicular component going round.

• (a)F is maximum when v and B are parallel to each other. — The exact opposite of the truth. Parallel means θ equals zero, so sin θ is zero and the magnetic force vanishes. A charge moving along the field lines is not deflected at all.
• (b)F is maximum when v and B are anti-parallel to each other. — Anti-parallel means θ equals 180 degrees, and sin 180 degrees is also zero. Reversing the direction of travel along the same line does not create a force where there was none.
• (d)The force F is independent of the angle between v and B. — The sin θ factor is precisely an angular dependence. The force runs continuously from zero when the motion is along the field to a maximum of qvB when the motion is across it — that variation is the heart of the phenomenon.

A magnetic field acts only on charge that is moving, and only on the part of that motion which cuts across the field lines. The Lorentz magnetic force F = q(v × B) is perpendicular to both the velocity and the field, so it changes the direction of motion but never the speed, and therefore does no work on the particle. The direction is given by Fleming's left-hand rule for a current, or by the right-hand rule for the cross product of v and B, with the sign reversed for a negative charge.

The item is really a test of whether a candidate remembers that the formula carries a sine and not a cosine. A quick physical check settles it without the algebra: a charge drifting straight along a field line has no way to tell which way is sideways, so there is no direction for a force to point, and the force must be zero. Turn the velocity across the field and the effect is at its strongest. Two familiar consequences hang on the same fact. Charged particles from the Sun spiral along the Earth's field lines and are funnelled towards the poles rather than striking the equator head-on, and a mass spectrometer or a cyclotron works by sending charges across a field so that they bend into circles of a radius that depends on their momentum and charge. Both are settled physics and nothing about the answer has moved since the 2023 exam. It is worth noting that the bank's subject label puts this item under Indian Economy, which is a tagging slip; the content is straightforward electromagnetism.

• The magnetic force on a moving charge is F = q(v × B), with magnitude F = qvB sin θ.
• The force is zero when the velocity is parallel or anti-parallel to the field, and maximum, equal to qvB, when the two are perpendicular.
• Because the force is always perpendicular to the velocity, it does no work — it changes direction but not speed.
• A charge moving across a uniform field travels in a circle; one moving obliquely traces a helix; one moving along the field goes straight.
• A stationary charge feels no magnetic force whatever the field strength, since v equals zero makes the whole product vanish.

• Remembering the formula with a cosine instead of a sine, which inverts every conclusion in the question.
• Thinking that anti-parallel motion must give a force in the opposite direction; both parallel and anti-parallel give zero.
• Assuming a magnetic field can speed a particle up. It bends the path but leaves the speed, and hence the kinetic energy, unchanged.

As a one-line statement item on when the force is maximum or zero, or as a numerical asking for the radius or period of the circular path of a charge in a field.

• practice — not a real PYQ

  A charged particle enters a uniform magnetic field moving exactly along the direction of the field. Its path will be

  • (a)a circle
  • (b)a helix
  • (c)a straight line
  • (d)a parabola

  Answer(c) a straight line — with the velocity parallel to the field, sin θ is zero, so the magnetic force is zero and the particle continues undeflected.

• practice — not a real PYQ

  The magnetic force acting on a charged particle moving in a magnetic field does no work on the particle because the force is

  • (a)always perpendicular to the velocity
  • (b)always parallel to the velocity
  • (c)independent of the charge
  • (d)independent of the speed

  Answer(a) always perpendicular to the velocity — a force at right angles to the motion changes direction only, so the speed and the kinetic energy stay the same.