Force between Parallel Currents — Revision Notes

⚡ 30-Second Revision

Magnetic Field (Straight Wire): — $B = \frac{\mu_0 I}{2\pi r}$ (Right-Hand Thumb Rule for direction)
Force on Current in Field: — $\vec{F} = I(\vec{L} \times \vec{B})$ (Fleming's Left-Hand Rule for direction)
Force per Unit Length (Parallel Wires): — $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$
Same Direction Currents: — Attract
Opposite Direction Currents: — Repel
$\mu_0$ (Permeability of Free Space): — $4\pi \times 10^{-7}\,\text{T}\cdot\text{m/A}$
Ampere Definition: — Based on force between parallel currents ($2 \times 10^{-7}\,\text{N/m}$ at $1,\text{m}$ separation for $1,\text{A}$ each).

2-Minute Revision

The force between parallel current-carrying wires is a direct consequence of magnetic fields. Each wire produces a magnetic field, and this field then exerts a force on the other wire. The magnitude of this force per unit length is given by $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$, where $I_1$ and $I_2$ are the currents, and $d$ is their separation.

A crucial aspect for NEET is the direction of this force: if currents flow in the same direction, the wires attract; if they flow in opposite directions, they repel. Remember to use the Right-Hand Thumb Rule for magnetic field direction and Fleming's Left-Hand Rule for force direction.

This phenomenon is also fundamental to the definition of the Ampere. Pay attention to units (especially distance in meters) and powers of 10 in numerical problems. For multiple wires, perform vector addition of individual forces.

5-Minute Revision

To master the force between parallel currents, start with the basics: a current-carrying wire generates a magnetic field ($B = \frac{\mu_0 I}{2\pi r}$, direction via Right-Hand Thumb Rule). A second current-carrying wire placed in this field experiences a Lorentz force ($\vec{F} = I(\vec{L} \times \vec{B})$, direction via Fleming's Left-Hand Rule). Combining these, the force per unit length between two parallel wires is $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$.

Key Directions:

Same-direction currents: — Attract. Imagine the fields between them partially cancelling, creating a 'lower pressure' zone that pulls them together.
Opposite-direction currents: — Repel. The fields between them add up, creating a 'higher pressure' zone that pushes them apart.

Example: Two wires, $I_1 = 4,\text{A}$ and $I_2 = 6,\text{A}$, are $0.2,\text{m}$ apart. If currents are in the same direction:

$$\frac{F}{L} = \frac{(4\pi \times 10^{-7}) \times 4 \times 6}{2\pi \times 0.2} = \frac{2 \times 10^{-7} \times 24}{0.2} = \frac{48 \times 10^{-7}}{0.2} = 240 \times 10^{-7} = 2.4 \times 10^{-5}\,\text{N/m}$$

The force is attractive.

NEET Focus:

1
Formula Application: — Direct calculation of $F/L$.
2
Direction: — Correctly identifying attraction or repulsion.
3
Multiple Wires: — Vector sum of forces. If three wires A, B, C are in a line, the force on B is $F_{BA} + F_{BC}$ (vector sum). Be careful with directions.
4
Ampere Definition: — Understand how the force formula defines the Ampere. Practice unit conversions (cm to m) and handling powers of 10. Avoid common mistakes like mixing up direction rules or calculation errors.

Prelims Revision Notes

1
Magnetic Field by Straight Wire: — A long, straight wire carrying current $I$ produces a magnetic field $B$ at a perpendicular distance $r$ given by $B = \frac{\mu_0 I}{2\pi r}$. The direction is found using the Right-Hand Thumb Rule (thumb = current, curled fingers = field).
2
Force on Current in Magnetic Field: — A conductor of length $L$ carrying current $I$ in a uniform magnetic field $\vec{B}$ experiences a force $\vec{F} = I(\vec{L} \times \vec{B})$. The magnitude is $F = I L B \sin\theta$, where $\theta$ is the angle between current and field. The direction is given by Fleming's Left-Hand Rule (Forefinger = Field, Middle finger = Current, Thumb = Force).
3
Force between Parallel Currents: — When two long, straight, parallel wires carry currents $I_1$ and $I_2$ separated by a distance $d$, the force per unit length ($F/L$) on each wire due to the other is:
$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$
where $\mu_0 = 4\pi \times 10^{-7}\,\text{T}\cdot\text{m/A}$ is the permeability of free space.
4
Direction of Force:

* Same Direction Currents: Wires attract each other. * Opposite Direction Currents: Wires repel each other.

1
Definition of Ampere: — One Ampere is defined as that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to $2 \times 10^{-7}$ Newton per metre of length.
2
Multiple Wires: — For a system of three or more parallel wires, the net force on any one wire is the vector sum of the forces exerted on it by each of the other wires. Carefully determine the direction of each individual force before summing.
3
Units and Constants: — Ensure all quantities are in SI units (current in Amperes, distance in meters, force in Newtons). Remember $\mu_0$ value.

Vyyuha Quick Recall

Same Direction Attracts, Opposite Direction Repels (SDAR, ODAR). For the formula, remember 'Mu-naught I-one I-two over two-pi-d' sounds like a rhythmic chant for $\frac{\mu_0 I_1 I_2}{2\pi d}$.