Problem 1873 - JEE Physics

Question

A long straight wire of radius "a" carries a steady current I the current is uniformly distributed across its cross-section. The ratio of the magnetic field at \(\mathrm{a} / 2\) and \(2 \mathrm{a}\) is (a) \((1 / 4)\) (b) 4 (c) 1 (d) \((1 / 2)\)

Step-by-Step Solution

Answer

The ratio of the magnetic field at \(a/2\) and \(2a\) is 4. The correct answer is (b).

Step 1: Use Ampère's Law to find the magnetic field expression.

According to Ampère's Law, the magnetic field at a distance \(r\) from the axis of a long straight wire is given by the formula: \[B(r) = \frac{\mu_0 I}{2 \pi r}\] Here, \(B(r)\) is the magnetic field at a distance \(r\), \(\mu_0\) is the permeability of free space, and \(I\) is the current carried by the wire. Now we can use this formula to find the magnetic fields at the distances \(a/2\) and \(2a\).

Step 2: Find the magnetic field at distance \(a/2\).

By substituting the distance \(a/2\) into the formula from step 1, we can find the magnetic field at this point: \[B\left(\frac{a}{2}\right) = \frac{\mu_0 I}{2 \pi \frac{a}{2}} = \frac{2\mu_0 I}{2 \pi a}\]

Step 3: Find the magnetic field at distance \(2a\).

By substituting the distance \(2a\) into the formula from step 1, we can find the magnetic field at this point: \[B(2a) = \frac{\mu_0 I}{2 \pi 2a} = \frac{\mu_0 I}{4 \pi a}\]

Step 4: Calculate the ratio of the magnetic fields.

Now we can compute the ratio of the magnetic fields at the two given distances: \[\frac{B\left(\frac{a}{2}\right)}{B(2a)} = \frac{\frac{2\mu_0 I}{2 \pi a}}{\frac{\mu_0 I}{4 \pi a}} = \frac{1}{1/4} = 4\] So, the ratio of the magnetic field at \(a/2\) and \(2a\) is 4. The correct answer is (b).

Key Concepts

Magnetic Field

The magnetic field is an invisible force field that surrounds a magnet or a current-carrying conductor like a wire. This field can exert a force on other magnets or moving charges in the vicinity, leading to phenomena such as the deflection of a compass needle or the operation of electric motors.
In the context of a current-carrying wire, the magnetic field lines form concentric circles around the wire. These lines are perpendicular to the direction of the electric current flowing through the wire.

• The strength of the magnetic field (denoted as \(B\)) depends on the distance from the wire. The farther away you are, the weaker the magnetic field becomes.
• Ampère's Law is a useful tool for calculating the magnetic field strength at different points around the wire. According to this law, the magnetic field at a distance \(r\) from a straight long wire is given by \(B(r) = \frac{\mu_0 I}{2 \pi r}\), where \(I\) is the current and \(\mu_0\) is the permeability of free space.

Understanding how magnetic fields behave around current-carrying wires is critical for designing electrical circuits and understanding electromagnetic interactions.

Current Distribution

In a conductor, current distribution refers to how the electric current spreads across the cross-section. For a straight wire carrying a steady current, the current distributes uniformly across its entire cross-sectional area.
Uniform current distribution implies that every part of the cross-section contributes equally to the magnetic field generated around the wire.

• Imagine slicing the wire into equal parts; each slice contributes proportionally to the total magnetic field.
• If the current were not uniformly distributed, the magnetic field would be uneven, complicating calculations of magnetic field strength.

This uniformity simplifies our calculations and is a key assumption when applying Ampère's Law in exercises like the one we are discussing. It allows us to predict the magnetic field at various distances accurately without having to adjust for variations in current density.

Permeability of Free Space

The permeability of free space, denoted by \(\mu_0\), is a fundamental physical constant that describes the ability of a vacuum to support the formation of a magnetic field. It's expressed in henries per meter (H/m) and has a fixed value of \(4\pi \times 10^{-7} \text{ H/m}\).
This constant is crucial in electromagnetic theory because it appears in fundamental equations like Ampère's Law, which relate current to magnetic field strength.

• In the equation \(B(r) = \frac{\mu_0 I}{2 \pi r}\), \(\mu_0\) serves as a proportionality constant, linking the magnetic field \(B\) to the current \(I\).
• \(\mu_0\) is a reflection of how "magnetically sensitive" a vacuum is. It's a baseline for measuring how other materials might affect magnetic fields, usually described by their relative permeability compared to \(\mu_0\).

Understanding \(\mu_0\) helps explain why magnetic fields behave the way they do in electrical and electromagnetic systems, forming the backbone of concepts in electromagnetism and circuit design.