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): \(B' = 2B\).