Two ideas combined
We have learnt that there exists a magnetic field due to a conductor carrying a current, which obeys the Biot-Savart law. Further, we have learnt that an external magnetic field will exert a force on a current-carrying conductor — this follows from the Lorentz force formula. Thus it is logical to expect that two current-carrying conductors placed near each other will exert (magnetic) forces on each other.
In the period 1820–25, Ampere studied the nature of this magnetic force and its dependence on the magnitude of the current, on the shape and size of the conductors, as well as the distances between the conductors. NCERT takes the simplest case — two long parallel conductors — to make the calculation transparent.
Wire a produces field Bₐ along wire b; the force on b due to this field points towards a. By symmetry b pulls a back equally — the wires attract.
Force on the second wire
Consider two long parallel conductors $a$ and $b$ separated by a distance $d$ and carrying (parallel) currents $I_a$ and $I_b$. The conductor $a$ produces the same magnetic field $B_a$ at all points along the conductor $b$. The right-hand rule tells us that the direction of this field is downwards (when the conductors are placed horizontally). Its magnitude, from the field of a long straight wire or from Ampere's circuital law, is:
$$B_a = \frac{\mu_0 I_a}{2\pi d}$$
The conductor $b$ carrying a current $I_b$ will experience a sideways force due to the field $B_a$. The direction of this force is towards the conductor $a$. We label this force $F_{ba}$, the force on a segment $L$ of $b$ due to $a$. Its magnitude, from the Lorentz force on a current-carrying wire $F = I\,L\,B$, is:
$$F_{ba} = I_b\, L\, B_a = \frac{\mu_0 I_a I_b L}{2\pi d}$$
It is of course possible to compute the force on $a$ due to $b$. From considerations similar to the above, the force $F_{ab}$ on a segment of length $L$ of $a$ due to the current in $b$ is equal in magnitude to $F_{ba}$, and directed towards $b$. Thus:
$$F_{ba} = -F_{ab}$$
Note that this is consistent with Newton's third law. Thus, at least for parallel conductors and steady currents, the Biot-Savart law and the Lorentz force yield results in accordance with Newton's third law.
Attract or repel
We have seen from above that currents flowing in the same direction attract each other. One can show that oppositely directed currents repel each other. NCERT states the rule plainly:
Parallel currents attract, and antiparallel currents repel.
This rule is the opposite of what we find in electrostatics. Like (same sign) charges repel each other, but like (parallel) currents attract each other. This single contrast is the most-tested point of the entire subtopic, so anchor it firmly before moving on.
Reverse one current and each force flips direction: the wires are pushed apart. Same geometry, same magnitude — only the sign of one current changes the outcome.
| Current directions | Field of one wire at the other | Force on each wire | Result |
|---|---|---|---|
| Same direction (parallel) | $B = \dfrac{\mu_0 I_1}{2\pi d}$ | Towards the other wire | Attraction |
| Opposite directions (antiparallel) | $B = \dfrac{\mu_0 I_1}{2\pi d}$ | Away from the other wire | Repulsion |
| Either case | — | $F_{ba} = -F_{ab}$ (equal & opposite) | Newton's third law holds |
Same direction ATTRACTS — opposite to the charge rule
Students transfer the electrostatic habit "like repels like" to currents and pick the wrong sign. For currents the logic inverts: same-direction (parallel) currents attract, and only opposite-direction (antiparallel) currents repel.
Like charges repel; like (parallel) currents attract. Memorise the contrast, not just one half of it.
This force is just the Lorentz force on a current-carrying wire applied to the field made by a neighbour. Revisit it to see where every term here comes from.
Force per unit length
Because the force $F_{ba}$ scales with the segment length $L$, the natural quantity is the force per unit length. Let $f_{ba}$ represent the magnitude of the force $F_{ba}$ per unit length. Then, dividing the force expression by $L$:
$$f_{ba} = \frac{\mu_0 I_a I_b}{2\pi d}$$
This is the central NEET formula. The force per metre grows with both currents and falls off inversely with the separation $d$ — not with $d^2$, since the field of a straight wire itself goes as $1/d$. The constant $\mu_0/2\pi = 2 \times 10^{-7}\ \text{T m A}^{-1}$ is what makes the numbers in the ampere definition come out cleanly.
Force per length, not total force
The formula $f = \mu_0 I_1 I_2 / 2\pi d$ gives newtons per metre. The total force on a finite segment is $F = fL = \mu_0 I_1 I_2 L / 2\pi d$. Read the question: if it asks for force per unit length, do not multiply by length, and if it gives a length, do not forget to.
$f = \dfrac{\mu_0 I_1 I_2}{2\pi d}$ (N m⁻¹) · $F = fL$ (N for a segment $L$).
Defining the ampere
The expression for $f_{ba}$ is used to define the ampere (A), which is one of the seven SI base units. Setting $I_a = I_b = 1\ \text{A}$ and $d = 1\ \text{m}$ gives $f = (2 \times 10^{-7})(1)(1)/1 = 2 \times 10^{-7}\ \text{N m}^{-1}$ — the exact figure quoted in the definition.
The ampere is the value of that steady current which, when maintained in each of the two very long, straight, parallel conductors of negligible cross-section, and placed one metre apart in vacuum, would produce on each of these conductors a force equal to $2 \times 10^{-7}$ newtons per metre of length.
This definition of the ampere was adopted in 1946. It is a theoretical definition. In practice, one must eliminate the effect of the earth's magnetic field and substitute very long wires by multiturn coils of appropriate geometries. An instrument called the current balance is used to measure this mechanical force.
The SI unit of charge, namely the coulomb, can now be defined in terms of the ampere: when a steady current of 1 A is set up in a conductor, the quantity of charge that flows through its cross-section in 1 s is one coulomb (1 C).
The 1946 SI definition of the ampere reads this geometry directly off $f = \mu_0 I_1 I_2 / 2\pi d$ with both currents 1 A and the spacing 1 m.
Quote the ampere definition exactly
The four numbers must all be present: two parallel conductors, 1 metre apart, in vacuum, 2 × 10⁻⁷ N per metre — for 1 A in each. Dropping "per metre of length" or swapping the exponent (it is 10⁻⁷, not 10⁻⁶) is the usual slip.
1 A in each + 1 m apart in vacuum ⇒ 2 × 10⁻⁷ N m⁻¹.
Force between two parallel currents in one screen
- One wire's field at the other: $B = \mu_0 I_1 / 2\pi d$.
- Force on a segment $L$: $F = \mu_0 I_1 I_2 L / 2\pi d$; force per unit length $f = \mu_0 I_1 I_2 / 2\pi d$.
- Parallel currents attract, antiparallel currents repel — opposite to the charge rule.
- Forces are equal and opposite, $F_{ba} = -F_{ab}$, consistent with Newton's third law.
- Ampere: 1 A in each of two wires 1 m apart in vacuum gives 2 × 10⁻⁷ N per metre (adopted 1946).
- Direction by right-hand rule for $B$, then Lorentz force $F = I\,L \times B$ on the second wire.