Two identical current carrying coaxial loops, carry current $I$ in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as $C$, (a) $\oint_c \vec{B}\cdot d\vec{l} = \mp 2\mu_0 I$. (b) the value of $\oint_c \vec{B}\cdot d\vec{l}$ is independent of sense of C. (c) there may be a point on C where B and dl are perpendicular. (d) B vanishes everywhere on C.

Two identical current carrying coaxial loops, carry current I in an opposite sense. A simple amperian loop pas

NCERT Exemplar · Q8

I

(a)

\oint_c \vec{B}\cdot d\vec{l} = \mp 2\mu_0 I

(b) the value of

\oint_c \vec{B}\cdot d\vec{l}

is independent of sense of C.

(d) B vanishes everywhere on C.

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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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The net current enclosed by $C$ is zero for either sense of traversal, so $\oint_C \vec{B}\cdot d\vec{l} = 0$ regardless of the sense of $C$ - matching options (b) and (c).

Setting up Ampere's law

\oint_C \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}

The two coaxial loops carry equal current $I$ but in opposite senses. The amperian loop $C$ threads through both of them exactly once, so

I_{\text{enc}} = I - I = 0 \qquad\Rightarrow\qquad \oint_C \vec{B}\cdot d\vec{l} = 0.

Checking (a)

(a) claims $\oint_C \vec{B}\cdot d\vec{l} = \mp 2\mu_0 I$ . This double-counts the current as if both loops contributed with the same sign; since they are opposite, the terms cancel to zero, not $2\mu_0 I$ . (a) is false.

Checking (b)

Reversing the sense in which $C$ is traversed flips the sign of $I_{\text{enc}}$ (from $I-I$ to $-(I-I)$ ), but since $I_{\text{enc}}=0$ either way, the value of the line integral is $0$ regardless of which sense $C$ is traversed. (b) is true.

Checking (c) …

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Q.Two identical current carrying coaxial loops, carry current $I$ in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as $C$ ,

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 …

Setting up Ampere's law

Checking (a)

Checking (b)

Checking (c) …

Method: Ampere's Law With Cancelling Currents -- Zero Circulation Doesn't Mean Zero Field

Steps

Q.Two identical current carrying coaxial loops, carry current III in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as CCC,

Setting up Ampere's law

Checking (a)

Checking (b)

Checking (c) …

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Q.Two identical current carrying coaxial loops, carry current $I$ in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as $C$ ,

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