Question

The strength of the magnetic field at a point \(\mathrm{y}\) near a long straight current carrying wire is \(\mathrm{B}\). The field at a distance \(\mathrm{y} / 2\) will be (a) B/2 (b) B \(/ 4\) (c) \(2 \mathrm{~B}\) (d) \(4 \mathrm{~B}\)

Short answer

The magnetic field strength at a distance \(y/2\) from the long straight current-carrying wire is twice the magnetic field strength at distance \(y\). Therefore, the correct answer is option (c): \(B' = 2B\).

Step-by-step solution

Identify the formula for magnetic field strength

The formula for the magnetic field strength (B) near a long straight current-carrying wire at a distance (r) is given by the following relation from the Biot-Savart law: \[B = \frac{\mu_0 I}{2\pi r}\] where \(\mu_0\) is the permeability of free space, \(I\) is the current through the wire, and \(r\) is the distance from the wire to the point where we wish to calculate the magnetic field strength.

Calculate the magnetic field strength at distance 'y'

We are asked to find the magnetic field strength at distance \(y\) and \(y/2\). Let's first find the magnetic field strength at distance \(y\), which is given as \(B\). We can derive the following equation from the formula above: \[B = \frac{\mu_0 I}{2\pi y}\]

Calculate the magnetic field strength at distance 'y/2'

Now let's calculate the magnetic field strength at distance \(y/2\), which we can denote as \(B'\). We'll substitute the distance as \(y/2\) in the formula for magnetic field strength: \[B' = \frac{\mu_0 I}{2\pi(\frac{y}{2})}\]

Find the relationship between B and B'

We need to find the relationship between \(B\) and \(B'\). Divide the equation of \(B'\) by the equation of \(B\): \[\frac{B'}{B} = \frac{\frac{\mu_0 I}{2\pi(\frac{y}{2})}}{\frac{\mu_0 I}{2\pi y}}\] We can see that \(\mu_0 I\) term cancels out in the numerator and denominator: \[\frac{B'}{B} = \frac{1}{\frac{1}{2}}\]

Solve the relationship between B and B'

Now, we can simplify the final equation: \[\frac{B'}{B} = 2\] So, we see that the magnetic field strength at distance \(y/2\) is twice the magnetic field strength at distance \(y\), which is option (c): \(B' = 2B\).

Key concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle that allows us to calculate the magnetic field produced by a current-carrying wire. It connects the magnetic field to the presence and movement of electrical charges within the wire. The mathematical representation of the Biot-Savart Law for a long straight wire is given by:\[B = \frac{\mu_0 I}{2\pi r}\]Where:

• \(B\) is the magnetic field produced in Tesla (T).
• \(\mu_0\) is the permeability of free space, approximately \(4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}\).
• \(I\) is the current through the wire in Amperes (A).
• \(r\) is the perpendicular distance from the wire to the point where the field is being calculated.

This formula highlights the inverse relationship between the magnetic field strength and distance from the wire, illustrating how the field weakens as you move away.

Long Straight Current-Carrying Wire

When calculating the magnetic field produced by a long, straight wire carrying current, several simplifications make the problem more manageable. The wire is considered to have infinite length, which allows us to assume the magnetic field has cylindrical symmetry around the wire. This simplification helps as the field does not vary along the length of the wire and only changes with distance from the wire. In practical problems, considering an infinitely long wire aids in ignoring edge effects, which occur at the ends of finite wires. This allows the use of the Biot-Savart Law or Ampère's Law to produce accurate results effectively. Recognizing these simplifications is key to understanding and efficiently solving problems involving long straight wires.

Permeability of Free Space

The permeability of free space, denoted as \(\mu_0\), is a constant that is crucial to calculating magnetic fields in a vacuum or air. It represents how much a material can "support" the formation of a magnetic field inside it. For free space, or vacuum, \(\mu_0\) is defined and fixed at:\[\mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}\]This constant acts as a proportionality factor in equations such as the Biot-Savart Law and is vital in various electromagnetism laws. Understanding \(\mu_0\) helps in comprehending how magnetic fields interact within different environments, although for most elementary problems, air or vacuum is assumed.

Relationship Between Magnetic Field and Distance

The relationship between the magnetic field strength \(B\) and the distance \(r\) from a current-carrying wire is important to understand. According to the Biot-Savart Law, the field strength is inversely proportional to the distance; this is mathematically represented as:\[ B \propto \frac{1}{r} \]This means as you move further from the wire, the magnetic field strength decreases. Specifically, if you halve the distance \(r\), the magnetic field \(B\) will double. This inverse relationship is crucial for solving problems involving the field around wires, as seen when comparing field strengths at different distances, like \(y\) and \(y/2\), where the field at \(y/2\) is found to be twice that at \(y\). Appreciating this concept can vastly improve understanding of how magnetic fields behave around current-carrying wires.