Q.A horizontal overhead power line carries a current of $90\ \text{A}$ in east to west direction. What is the magnitude and direction of the magnetic field due to the current $1.5\ \text{m}$ below the line?

The key idea is the magnetic field around a long straight current-carrying conductor, given by Ampere’s law.

Step 1 – Formula: For an infinitely long straight wire, the magnitude of the magnetic field at a perpendicular distance $r$ is

Step 2 – Plug in values:

$I = 90\ \text{A}$, $r = 1.5\ \text{m}$, $\mu_0 = 4\pi \times 10^{-7}\ \text{T m/A}$

Step 3 – Calculate:

$B = \frac{180 \times 10^{-7}}{1.5} = 120 \times 10^{-7} = 1.2 \times 10^{-5}\ \text{T}$ …

Why magnetic force balance? — The core idea

This problem is not about forces on charges, but about the magnetic field produced by a current. The key physics: a long straight wire carrying current $I$ creates a magnetic field that circles around it. The field’s magnitude at a perpendicular distance $r$ is given by Ampere’s law (or Biot–Savart):

The direction is found using the right-hand thumb rule: point your thumb along the current, and your fingers curl in the direction of the magnetic field lines. For a horizontal wire running east–west, the field below the wire will be purely horizontal — but which way?

Let’s work it out step by step.

1. Identify the given data

   Current $I = 90\ \text{A}$, direction: east to west.

   Distance below the wire: $r = 1.5\ \text{m}$.

   Permeability of free space: $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$.

2. Apply the formula for magnetic field magnitude

   For an infinitely long straight wire (a very good approximation here):

Substitute values:

The $\pi$ cancels:

Simplify stepwise:

3. Determine the direction using the right-hand thumb rule Point your right thumb from east to west (the current direction). Now look at a point below the wire. Your fingers will curl toward the south at that location. …

Method: Right-Hand Thumb Rule & Biot–Savart Law for a Long Straight Conductor

This problem uses the standard formula for the magnetic field around an infinitely long straight current-carrying wire.

Step 1: Identify the given data

• Current, $I = 90\ \text{A}$ (east to west)
• Perpendicular distance from wire, $r = 1.5\ \text{m}$
• We need $B$ at a point below the wire.

Step 2: Recall the formula

For a long straight conductor, the magnitude of the magnetic field at a distance $r$ is:

where

$\mu_0 = 4\pi \times 10^{-7}\ \text{T m A}^{-1}$ (permeability of free space).

Step 3: Substitute and calculate magnitude

Cancel $\pi$:

Simplify step-by-step:

$\boxed{B = 1.2 \times 10^{-5}\ \text{T}}$ …

Here are the most common mistakes students make on this classic magnetic field problem, along with how to avoid each one.

Mistake 1: Using the Wrong Formula (Force vs. Field)

• The Mistake: Students see "current" and "wire" and immediately write $F = BIL$ or $F = qvB$, trying to calculate a force. The question asks for the magnetic field ($B$) due to the wire, not the force on another charge or wire.
• How to Avoid: Read the question stem carefully. If it asks for "magnetic field due to the current," you need the Biot-Savart law for a long straight wire:

Only use force formulas if the problem asks for the force on a second wire or a moving charge.

Mistake 2: Incorrect Distance ($r$) in the Denominator

• The Mistake: Plugging in $r = 1.5\ \text{m}$ directly but forgetting that $r$ is the perpendicular distance from the wire to the point. In this case, "below the line" means exactly perpendicular, so $r = 1.5\ \text{m}$ is correct. However, students often confuse this with the length of the wire or the distance along the wire.
• How to Avoid: Always draw a quick diagram. The wire is horizontal. The point is directly below it. The shortest distance from the wire to the point is the vertical drop — that’s your $r$. If the point were off to the side, you’d need the perpendicular distance, not the slant distance.

Mistake 3: Forgetting the Constant $\mu_0$

• The Mistake: Using $\mu_0 = 4\pi \times 10^{-7}$ incorrectly, or worse, omitting it entirely and just computing $\frac{I}{2\pi r}$.
• How to Avoid: Memorize the standard value:

Then simplify the formula:

Notice the $\pi$ cancels — this is a common simplification that saves time and reduces error.

Mistake 4: Direction Error (Right-Hand Thumb Rule)

• The Mistake: Getting the direction of the magnetic field lines wrong. Common errors include pointing the thumb in the direction of the current but then confusing which way the fingers curl.
• How to Avoid: Use the Right-Hand Thumb Rule:
  1. Point your right thumb in the direction of the current (East to West).
  2. Your curled fingers show the direction of the magnetic field lines.
  • At a point below the wire: Your fingers will be curling into the page (or screen) at that point. …