Q.Two long and parallel straight wires A and B carrying currents of $8.0\ \text{A}$ and $5.0\ \text{A}$ in the same direction are separated by a distance of $4.0\ \text{cm}$. Estimate the force on a $10\ \text{cm}$ section of wire A.

Concept: Magnetic Force Between Parallel Currents

Parallel currents in the same direction attract each other. The force per unit length between two long straight wires is given by:

Step 1 – Identify the given values

$I_1 = 8.0\ \text{A}$, $I_2 = 5.0\ \text{A}$, $d = 4.0\ \text{cm} = 0.040\ \text{m}$, $L = 10\ \text{cm} = 0.10\ \text{m}$, and $\mu_0 = 4\pi \times 10^{-7}\ \text{N/A}^2$.

Step 2 – Calculate force per unit length

Simplify: …

Concept and Intuition: Magnetic Force Balance

When two wires carry current, each wire creates a magnetic field around it. The other wire, sitting in that field, experiences a magnetic force. For long straight parallel wires, the field from one wire is circular and its magnitude at the location of the other wire is uniform along the length. The direction of the force depends on whether the currents are in the same direction (attraction) or opposite (repulsion). Here, both currents flow the same way, so the wires pull toward each other.

The key formula comes from Ampere’s law and the Lorentz force: the force per unit length between two parallel wires carrying currents $I_1$ and $I_2$, separated by distance $r$, is

where $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$ is the permeability of free space. This is a standard result you should remember for exams.

Step-by-Step Solution

1. Identify the given quantities.

Current in wire A: $I_A = 8.0\ \text{A}$

Current in wire B: $I_B = 5.0\ \text{A}$

Separation: $r = 4.0\ \text{cm} = 0.040\ \text{m}$

Length of section considered: $L = 10\ \text{cm} = 0.10\ \text{m}$

Both currents are in the same direction.

2. Write the force per unit length formula.

The magnitude of the force per unit length on either wire is

3. Substitute the values.

Plug in $\mu_0 = 4\pi \times 10^{-7}$, $I_A = 8.0$, $I_B = 5.0$, $r = 0.040$:

Notice that $\pi$ cancels:

Simplify step by step: …

Method: Force Between Parallel Current-Carrying Wires (Ampère’s Force Law)

This method uses the magnetic field created by one wire and the Lorentz force experienced by the other wire placed in that field.

Steps

1. Identify the known quantities

   • Current in wire A: $I_A = 8.0\ \text{A}$
   • Current in wire B: $I_B = 5.0\ \text{A}$
   • Separation between wires: $r = 4.0\ \text{cm} = 0.04\ \text{m}$
   • Length of section considered on wire A: $L = 10\ \text{cm} = 0.10\ \text{m}$
   • Permeability of free space: $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$

2. Recall the formula for force per unit length between two parallel wires

   The magnetic force per unit length on either wire is given by:

3. Substitute the values

4. Simplify step-by-step
   • Cancel $\pi$:

• Numerator: $4 \times 10^{-7} \times 40 = 1.6 \times 10^{-5}$
• Denominator: $2 \times 0.04 = 0.08$
• So: …

Here are the common mistakes students make when solving this Magnetic Force Balance problem, and how to avoid each.

1. Forgetting the Formula for Force Between Parallel Wires

Mistake:

Students use $F = qvB$ or $F = BIL$ without the correct expression for the field from the other wire.

How to avoid:

Always start with the force per unit length between two parallel currents:

where:

• $\mu_0 = 4\pi \times 10^{-7}\ \text{T m/A}$
• $I_1, I_2$ are the currents
• $d$ is the separation between wires

Then multiply by the length $L$ of the section considered.

2. Using Wrong Units for Distance

Mistake:

Plugging $d = 4.0$ (in cm) directly into the formula without converting to metres.

How to avoid:

Always convert cm → m before substituting:

Similarly, length $L = 10\ \text{cm} = 0.10\ \text{m}$.

3. Confusing Attraction vs Repulsion Direction

Mistake:

Stating the force is repulsive when currents are in the same direction.

How to avoid:

Remember the rule:

• Same direction → wires attract
• Opposite direction → wires repel

Here, both currents are $8.0\ \text{A}$ and $5.0\ \text{A}$ in the same direction, so the force on wire A is attractive toward wire B.

4. Forgetting to Multiply by Length After Finding Force per Unit Length

Mistake:

Stopping at $\frac{F}{L}$ and reporting that as the final answer.

How to avoid:

Always check:

• Did the question ask for force (in N) or force per unit length (in N/m)?
• Here it asks for force on a 10 cm section, so multiply:

5. Sign Errors or Omitting $\mu_0$ Value

Mistake:

Using $\mu_0 = 4\pi \times 10^{-7}$ incorrectly, or forgetting it entirely.

How to avoid:

Memorise $\mu_0 = 4\pi \times 10^{-7}$ exactly. Notice that $4\pi$ cancels with the $4\pi$ in the denominator — a common simplification: …