Understanding the Problem: Magnetic Field in a Circular Loop

The question asks us to find the amount of current flowing through a wire that is bent into a circular loop. We are given the total length of the wire, L, and the magnetic field strength, B, measured at the exact center of the loop. We need to use the relationship between the current, the loop's dimensions, and the magnetic field it produces at its center to solve for the current.

Relating Wire Length to Loop Radius

When a wire of length L is bent into a circular loop, the total length of the wire becomes the circumference of the circle. Let the radius of the circular loop be r.

The formula for the circumference of a circle is $C = 2\pi r$.

In this case, the circumference C is equal to the length of the wire L.

So, $L = 2\pi r$.

From this equation, we can find the radius r in terms of L:

$\displaystyle r = \frac{L}{2\pi}$

Magnetic Field at the Center of a Circular Loop

The formula for the magnetic field induction B at the center of a single circular loop carrying a current I is given by:

$\displaystyle B = \frac{{\mu _0 I}}{{2r}}$

where $\mu_0$ is the permeability of free space.

Calculating the Current Passing Through the Loop

Now we substitute the expression for r (which is $\frac{L}{2\pi}$) into the formula for B:

$\displaystyle B = \frac{{\mu _0 I}}{{2\left( {\frac{L}{{2\pi }}} \right)}}$

Simplify the denominator:

$\displaystyle B = \frac{{\mu _0 I}}{{\frac{{2L}}{{2\pi }}}} = \frac{{\mu _0 I}}{{\frac{L}{\pi }}}$

Now, rearrange the equation to solve for the current I:

$\displaystyle B \times \frac{L}{\pi } = \mu _0 I$

$\displaystyle I = \frac{{B\frac{L}{\pi }}}{{\mu _0}}$

$\displaystyle I = \frac{{BL}}{{\mu _0 \pi }}$

Thus, the current passing through the loop is $\frac{{BL}}{{{\mu _0}\pi }}$.

Verification with Options

Let's compare our calculated current with the given options:

• Option 1: $\frac{{B\pi L}}{{{\mu _0}}}$
• Option 2: $\frac{{BL}}{{{\mu _0}\pi }}$
• Option 3: $\frac{{B\pi }}{{L{\mu _0}}}$
• Option 4: $\frac{{BL}}{\pi }$

Our result $\frac{{BL}}{{{\mu _0}\pi }}$ matches Option 2.

Revision Table: Magnetic Field Concepts

Additional Information: Magnetic Fields from Currents

The magnetic field produced by a current depends on the shape of the conductor and the distance from it. Here are a few important cases:

• Straight Wire: The magnetic field lines around a long straight current-carrying wire are concentric circles. The magnitude of the field at a distance 'd' from the wire is given by $B = \frac{{\mu _0 I}}{{2\pi d}}$.
• Solenoid: A solenoid is a coil of wire wound into a tightly packed helix. The magnetic field inside a long solenoid is nearly uniform and parallel to the axis. Its magnitude is given by $B = \mu_0 n I$, where n is the number of turns per unit length ($n = N/L_{solenoid}$).
• Toroid: A toroid is a solenoid bent into a circle. The magnetic field is contained within the toroid and is given by $B = \frac{{\mu _0 N I}}{{2\pi R}}$, where N is the total number of turns and R is the average radius of the toroid.
• Circular Loop with N turns: If the circular loop consists of N turns instead of just one, the magnetic field at the center is N times the field of a single loop: $B = \frac{{\mu _0 N I}}{{2r}}$. In our problem, the wire is bent into a single loop (N=1), so the formula for a single loop is used.

Understanding these basic configurations helps in solving more complex problems involving magnetic fields created by electric currents.