Q.The horizontal component of the earth's magnetic field at a certain place is $3.0 \times 10^{-5}\ \text{T}$ and the direction of the field is from the geographic south to the geographic north. A very long straight conductor is carrying a steady current of $1\ \text{A}$. What is the force per unit length on it when it is placed on a horizontal table and the direction of the current is

Concept: force on a current-carrying conductor, $\vec F=I\,\vec L\times\vec B$, so $F/L=IB\sin\theta$.

Given $B=3.0\times10^{-5}\ \text{T}$ (south $\to$ north, horizontal) and $I=1\ \text{A}$.

(a) Current east $\to$ west. The current (west) is perpendicular to the field (north), $\theta=90^\circ$:

$\frac{F}{L}=IB\sin90^\circ=(1)(3.0\times10^{-5})=3.0\times10^{-5}\ \text{N/m}.$ …

Principle

A straight wire of length $L$ carrying current $I$ in a field $\vec B$ feels $\vec F=I\,\vec L\times\vec B$, of magnitude $F=ILB\sin\theta$, with $\theta$ the angle between the current and the field. Only the given horizontal component $B=3.0\times10^{-5}\ \text{T}$ (directed south $\to$ north) matters here.

Set up axes: east $=\hat x$, north $=\hat y$, vertically up $=\hat z$. Then $\vec B=3.0\times10^{-5}\,\hat y\ \text{T}$, and $I=1\ \text{A}$.

Part (a): current east to west

West is $-\hat x$, perpendicular to the north field, so $\theta=90^\circ$:

Direction from the cross product ($\hat L=-\hat x$):

i.e. along $-\hat z$ — vertically downward. …

Method: Magnetic Force on a Current-Carrying Conductor (Fleming’s Left-Hand Rule)

This problem uses the formula for the magnetic force on a straight current-carrying wire in a uniform magnetic field:

Where:

• $F$ = force on the conductor (N)
• $I$ = current (A)
• $L$ = length of conductor (m)
• $B$ = magnetic field strength (T)
• $\theta$ = angle between the direction of current and the magnetic field

We need force per unit length, so:

Given Data

• $B = 3.0 \times 10^{-5}\ \text{T}$ (horizontal, from geographic south → north)
• $I = 1\ \text{A}$
• Conductor placed on a horizontal table

Step-by-step for each case

(a) Current direction: East to West

1. Identify the angle $\theta$

   • Magnetic field direction: South → North
   • Current direction: East → West
   • These are perpendicular (90° apart horizontally)

2. Apply the formula

3. Result

Direction: Using Fleming’s Left-Hand Rule, the force is vertically upward (out of the table).

(b) Current direction: South to North

1. Identify the angle $\theta$
   • Magnetic field direction: South → North
   • Current direction: South → North …

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

Mistake 1: Forgetting the Direction of the Earth’s Horizontal Field

The Mistake

Students assume the horizontal component points from geographic north to south (like a compass needle). But the problem explicitly says: from geographic south to geographic north.

Why it matters

The direction of the magnetic field determines the sign and direction of the force via the cross product $\vec{F} = I (\vec{L} \times \vec{B})$.

How to Avoid

• Read the problem statement carefully — do not rely on memory.
• Draw a clear diagram: label geographic N, S, E, W. Mark the field arrow pointing from S to N (i.e., towards geographic north).
• For a wire on a horizontal table, the field is horizontal, so the force will be vertical (up or down).

Mistake 2: Using the Wrong Formula or Forgetting the Cross Product

The Mistake

Students write $F = ILB$ without considering the angle between current and field. They forget that the force per unit length is:

where $\theta$ is the angle between the current direction and the magnetic field direction.

How to Avoid

• Always start with the vector form: $\vec{F} = I (\vec{L} \times \vec{B})$.
• Identify $\theta$ explicitly:
  • For part (a): current east to west, field south to north → angle = $90^\circ$ → $\sin 90^\circ = 1$.
  • For part (b): current south to north, field south to north → angle = $0^\circ$ → $\sin 0^\circ = 0$.

Mistake 3: Getting the Direction of Force Wrong (Especially Part a)

The Mistake

Students apply Fleming’s left-hand rule incorrectly or forget that the force is perpendicular to both current and field. For part (a), they often guess the force is horizontal.

Correct Reasoning

• Current: east to west (say along $-\hat{x}$).
• Field: south to north (say along $+\hat{y}$).
• Cross product: $\vec{L} \times \vec{B}$ → $(-\hat{x}) \times (+\hat{y}) = -\hat{z}$ (downward).
• So the force per unit length is vertically downward.

How to Avoid

• Use the right-hand rule for cross product: point fingers along current, curl them toward field, thumb gives force direction.
• Draw a 3D coordinate system on your paper:
  • $x$: east-west
  • $y$: south-north
  • $z$: vertical (up = $+\hat{z}$)
• Practice with both Fleming’s rule and the cross product until they agree.

Mistake 4: Calculating Force When It Should Be Zero (Part b)

The Mistake

Students plug numbers into $F/L = I B$ for part (b) and get a non-zero answer, forgetting that parallel current and field give zero force.

Correct Reasoning

• Current direction: south to north (same as field).
• $\theta = 0^\circ$, $\sin 0^\circ = 0$ → $F/L = 0$.

How to Avoid …