A charged particle would continue to move with a constant velocity in a region wherein, (a) E = 0,B ≠ 0. (b) E

NCERT Exemplar · Q10

Q. A charged particle would continue to move with a constant velocity in a region wherein,

(a)

E = 0,\ B \neq 0

(b)

E \neq 0,\ B \neq 0

(c)

E \neq 0,\ B = 0

(d)

E = 0,\ B = 0

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Concept understanding — Magnetic Force Balance

Magnetic Force Balance

When a current-carrying wire or coil sits in a magnetic field, it feels a force $F = BIL\sin\theta$ (or, for a point charge, $F=qvB\sin\theta$ ). On its own that force just pushes the conductor - but in many real situations the push is deliberately set up to CANCEL another force, so the whole system sits in equilibrium. That equilibrium condition - magnetic force balanced against weight, against another wire's magnetic force, or against a mechanical counterweight - is what "magnetic force balance" means, and it is also historically how the ampere itself was defined.

The balance condition

Whenever a conductor is in equilibrium under a magnetic force and one other force, the two must be equal and opposite:

BIL\sin\theta = F_{\text{other}}

Solving this equation for whichever quantity is unknown ( $B$ , $I$ , $L$ , or the other force) is the entire skill in this class of problem - the only new step, beyond the force law itself, is correctly identifying what the magnetic force is opposing.

Case 1: a wire suspended against gravity

A straight horizontal wire of mass $m$ and length $l$ , carrying current $I$ , can be held up ("floated") in mid-air by a horizontal magnetic field perpendicular to it. The upward magnetic force must equal the downward weight:

BIl = mg \quad\Longrightarrow\quad B=\frac{mg}{Il}

For example, a $200\,\text{g}$ , $1.5\,\text{m}$ wire carrying $2\,\text{A}$ needs $B=\dfrac{(0.2)(9.8)}{(2)(1.5)}\approx0.65\,\text{T}$ to stay suspended.

Case 2: two wires balancing each other

Two long parallel wires carrying currents $I_1,I_2$ exert a force per unit length on each other of $\dfrac{\mu_0 I_1I_2}{2\pi d}$ (attractive if the currents run the same way, repulsive if opposite). If one wire is free to move, this magnetic force can itself balance that wire's weight:

\frac{\mu_0 I^2 L}{2\pi h}=mg \quad\Longrightarrow\quad h=\frac{\mu_0 I^2 L}{2\pi mg}

This is exactly how a "current balance" apparatus works, and historically it is how the ampere was defined: the current that, flowing in two infinite parallel wires one metre apart, produces a force of exactly $2\times10^{-7}\,\text{N}$ per metre of length.

Case 3: balancing on a beam

A current-carrying coil arm hanging from one pan of a beam balance feels an extra force $F=NBIl$ when only that arm sits in an external field. Re-balancing the beam means adding a mass $m$ so that $mg = NBIl$ .

Watch out

Always check which length enters the formula - for a coil of $N$ turns the force multiplies by $N$ ; for a single suspended straight wire it doesn't. …

Why this formula?

Magnetic Force Balance: Why the Key Formulas Hold

The Magnetic Force Balance describes when the magnetic force on a charged particle or current-carrying conductor is exactly balanced by another force (gravity, electric force, or tension). Let's build the reasoning step-by-step.

1. The Core Idea: What Does "Balance" Mean?

A force balance means the net force on an object is zero:

\vec{F}_{\text{net}} = 0

For magnetic forces we use the Lorentz force law:

On a moving charge: $\vec{F}_m = q (\vec{v} \times \vec{B})$
On a current-carrying wire: $\vec{F}_m = I (\vec{L} \times \vec{B})$

When this is balanced by another force (say gravity $\vec{F}_g = m\vec{g}$ ):

\vec{F}_m + \vec{F}_{\text{other}} = 0

2. Case 1: Charged Particle in Crossed Fields (Velocity Selector)

A charged particle moves perpendicular to both electric field $\vec{E}$ and magnetic field $\vec{B}$ .

Electric force: $\vec{F}_e = q\vec{E}$ (along $\vec{E}$ )
Magnetic force: $\vec{F}_m = q(\vec{v} \times \vec{B})$ (perpendicular to both $\vec{v}$ and $\vec{B}$ )

For straight-line motion (no deflection), the two forces must cancel:

qE = qvB \quad \Rightarrow \quad \boxed{v = \frac{E}{B}}

Key insight: Only particles with this exact speed pass undeflected — this is how velocity selectors work in mass spectrometers.

3. Case 2: Current-Carrying Wire Balanced by Gravity

A horizontal wire carrying current $I$ sits in a perpendicular magnetic field $\vec{B}$ , suspended by strings.

The magnetic force on a straight wire is $F_m = I L B \sin\theta$ ; for a wire perpendicular to the field ( $\theta = 90^\circ$ ), $F_m = ILB$ . Setting this equal to the weight $F_g = mg$ for equilibrium:

\boxed{I L B = mg}

Key insight: This balance lets you measure $B$ if $I$ , $L$ , and $m$ are known — the principle behind a current balance experiment.

4. Case 3: Circular Motion of a Charged Particle …

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Constant velocity needs zero net Lorentz force, $q(\vec{E}+\vec{v}\times\vec{B})=0$ . That is possible for (a) $E=0,B\neq0$ with $\vec{v}\parallel\vec{B}$ ; for (b) $E\neq0,B\neq0$ with $\vec{E}=-\vec{v}\times\vec{B}$ ; and for (d) $E=0,B=0$ trivially. It is impossible for (c) $E\neq0,B=0$ .

Condition for constant velocity

Constant velocity means zero acceleration, hence zero net force. The only force acting is the Lorentz force, so

q(\vec{E}+\vec{v}\times\vec{B})=0\quad\Longrightarrow\quad \vec{E}+\vec{v}\times\vec{B}=0.

(a) $E=0,\ B\neq0$ . The condition reduces to $\vec{v}\times\vec{B}=0$ , satisfied when $\vec{v}$ is parallel (or anti-parallel) to $\vec{B}$ . The magnetic force is then zero and the velocity stays unchanged. Possible.

(b) $E\neq0,\ B\neq0$ . Possible when the electric and magnetic forces exactly cancel: $\vec{E}=-\vec{v}\times\vec{B}$ (with $\vec{E}\perp\vec{B}$ and speed $v=E/B$ ). This is precisely the working principle of a velocity selector. Possible. …

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Q.A charged particle would continue to move with a constant velocity in a region wherein,

Concept understanding — Magnetic Force Balance

Magnetic Force Balance

The balance condition

Case 1: a wire suspended against gravity

Case 2: two wires balancing each other

Case 3: balancing on a beam

Why this formula?

Magnetic Force Balance: Why the Key Formulas Hold

1. The Core Idea: What Does "Balance" Mean?

2. Case 1: Charged Particle in Crossed Fields (Velocity Selector)

3. Case 2: Current-Carrying Wire Balanced by Gravity

4. Case 3: Circular Motion of a Charged Particle …

Condition for constant velocity

Method: Testing Whether a Charged Particle Can Move With Constant Velocity

Steps

Q.A charged particle would continue to move with a constant velocity in a region wherein,

Condition for constant velocity

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