Problem 1874 - JEE Physics

Question

At a distance of \(10 \mathrm{~cm}\) from a long straight wire carrying current, the magnetic field is \(4 \times 10^{-2}\). At the distance of \(40 \mathrm{~cm}\), the magnetic field will be Tesla. (a) \(1 \times 10^{-2}\) (b) \(2 \times \overline{10^{-2}}\) (c) \(8 \times 10^{-2}\) (d) \(16 \times 10^{-2}\)

Step-by-Step Solution

Answer

The short answer is: (a) \(1 \times 10^{-2}\)

Step 1: Write down the formula for the magnetic field due to a long straight wire carrying current

The formula for the magnetic field B at a distance r from a long straight wire carrying current I is given by: \[ B = \frac{\mu_0 I}{2 \pi r} \] where \(\mu_0\) is the permeability of free space and has a value of \(4 \pi \times 10^{-7} Tm/A\).

Step 2: Calculate the current I in the wire

We are given the magnetic field at a distance of 10 cm, which is \(B_1 = 4 \times 10^{-2} T\). Using the formula and solving for I, we get: \[ I = \frac{2 \pi r_1 B_1}{\mu_0} \] where \(r_1 = 10 \times 10^{-2} m\) (converting cm to m). Now, substitute the given values and calculate the current I: \[ I = \frac{2 \pi (10 \times 10^{-2})(4 \times 10^{-2})}{(4 \pi \times 10^{-7})} \] \[ I = 2 \times 10^4 A \]

Step 3: Calculate the magnetic field at a distance of 40 cm

Now we have the current I in the wire, and we want to find the magnetic field at a distance of 40 cm, which is \(r_2 = 40 \times 10^{-2} m\). Using the formula and substituting the given values, we get: \[ B_2 = \frac{\mu_0 I}{2 \pi r_2} \] \[ B_2 = \frac{4 \pi \times 10^{-7} \times 2 \times 10^4}{2 \pi (40 \times 10^{-2})} \] \[ B_2 = 1 \times 10^{-2} T \] So the magnetic field at a distance of 40 cm from the wire is 1 × 10⁻² T. The correct answer is: (a) \(1 \times 10^{-2}\)

Key Concepts

Long Straight Wire

A long straight wire is an idealization used in physics to make the calculations of magnetic fields more manageable. Imagine a wire that extends infinitely in both directions. This helps us focus on the magnetic field at a certain distance without worrying about edge effects that occur with finite wires. When a current flows through this long straight wire, it creates a magnetic field around it.
This magnetic field forms concentric circles perpendicular to the wire.

• The direction of the magnetic field can be determined using the "right-hand rule." Point your thumb in the direction of the current, and your fingers will curl in the direction of the magnetic field.
• The magnitude of this magnetic field decreases as the distance from the wire increases.
• A key point in applying the long straight wire model is to assume the wire is sufficiently long so the magnetic field becomes consistent along the length considered.

Understanding these properties is essential when using the long straight wire assumption in calculating magnetic fields.

Permeability of Free Space

The permeability of free space, denoted as \( \mu_0 \), is a fundamental physical constant that characterizes the magnetic property of free space or vacuum. It is the measure of the ability of free space to support a magnetic field. In reality, this constant appears in the formula for magnetic fields created by electric currents.
The value of the permeability of free space is given by:

• \( \mu_0 = 4 \pi \times 10^{-7} \) Tesla meter per Ampere (Tm/A).
• This constant permits the derivation of the magnetic field formula for different configurations, such as solenoids or straight wires.
• In SI units, \( \mu_0 \) relates magnetic field strength and electric current.

By understanding \( \mu_0 \), it's possible to comprehend how magnetic fields interact with electric currents in a vacuum or air, which serves as a baseline for understanding more complex media.

Magnetic Field Formula

The magnetic field formula is crucial for calculating the magnetic field generated by a current-carrying conductor, such as a long straight wire. This formula quantitatively relates the magnetic field (\( B \)) to the current (\( I \)), permeability of free space (\( \mu_0 \)), and the distance from the wire (\( r \)). The formula is expressed as:\[B = \frac{\mu_0 I}{2 \pi r}\]Here’s a breakdown of the formula's components:

• \( B \) represents the magnetic field strength at a distance \( r \) from the wire.
• \( I \) is the current flowing through the wire.
• \( \mu_0 \) is the permeability of free space, determining how easily a magnetic field forms in the vacuum surrounding the wire.
• \( r \) is the perpendicular distance from the wire to the point where the magnetic field is being calculated.

This formula shows that the magnetic field strength decreases as the distance from the wire increases. It highlights how current and distance influence the field's strength, helping us understand phenomena in electromagnetism, such as how magnetic fields behave around current-carrying wires.