A flat circular coil having 10 loops of wire has a diameter...

A flat circular coil having 10 loops of wire has a diameter of $2.0 \mathrm{~cm}$ and carries a current of $0.50$ A. It is mounted inside a long solenoid immersed in air, that has 200 loops on its $25-\mathrm{cm}$ length. The current in the solenoid is $2.4$ A. Compute the torque required to hold the coil with its central axis perpendicular to that of the solenoid. Let the subscripts $s$ and $c$ refer to the solenoid and coil, respectively. Then $$ \tau=N_{c} I_{\mathrm{c}} A_{\mathrm{c}} B_{8} \sin 90^{\circ} $$ But $B_{s}=\mu_{0} n I_{s}=\mu_{0}\left(N_{s} / L_{s}\right) I_{s}$, which gives $$ \begin{aligned} \tau &=\frac{\mu_{0} N_{c} N_{s} I_{c} I_{s}\left(\pi r_{c}^{2}\right)}{L_{s}} \\ &=\frac{\left(4 \pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}\right)(10)(200)(0.50 \mathrm{~A})(2.4 \mathrm{~A}) \pi(0.010 \mathrm{~m})^{2}}{0.25 \mathrm{~m}} \\ &=3.8 \times 10^{-6} \mathrm{~N} \cdot \mathrm{m} \end{aligned} $$

A flat circular coil having 10 loops of wire has a diameter of $2.0 \mathrm{~cm}$ and carries a current of $0.50$
A. It is mounted inside a long solenoid immersed in air, that has 200 loops on its $25-\mathrm{cm}$ length. The current in the solenoid is $2.4$ A. Compute the torque required to hold the coil with its central axis perpendicular to that of the solenoid. Let the subscripts $s$ and $c$ refer to the solenoid and coil, respectively. Then
$$
\tau=N_{c} I_{\mathrm{c}} A_{\mathrm{c}} B_{8} \sin 90^{\circ}
$$
But $B_{s}=\mu_{0} n I_{s}=\mu_{0}\left(N_{s} / L_{s}\right) I_{s}$, which gives
$$
\begin{aligned}
\tau &=\frac{\mu_{0} N_{c} N_{s} I_{c} I_{s}\left(\pi r_{c}^{2}\right)}{L_{s}} \\
&=\frac{\left(4 \pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}\right)(10)(200)(0.50 \mathrm{~A})(2.4 \mathrm{~A}) \pi(0.010 \mathrm{~m})^{2}}{0.25 \mathrm{~m}} \\
&=3.8 \times 10^{-6} \mathrm{~N} \cdot \mathrm{m}
\end{aligned}
$$

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Key Concepts

Magnetic Field of a Solenoid

In a long solenoid, the magnetic field is nearly uniform and is calculated using the formula B = ?? (N/L) I, where ?? is the permeability of free space, N is the number of turns, L is the solenoid’s length, and I is the current. This concept is fundamental in understanding how current configurations can generate a uniform magnetic environment, which is essential for analyzing interactions with magnetic dipoles or coils placed within the solenoid.

Magnetic Dipole Moment

The magnetic dipole moment of a current-carrying coil is defined by m = N I A, where N is the number of loops, I is the current through the coil, and A is the area of a single loop. This parameter characterizes the strength and orientation of the coil’s magnetic field and is crucial for predicting the coil's behavior in an external magnetic field.

Torque on a Magnetic Dipole

A magnetic dipole placed in an external magnetic field experiences a torque given by ? = m × B. For the maximum torque, when the dipole moment is perpendicular to the field, this simplifies to ? = mB. This principle is important in applications such as electric motors and magnetic sensors, as it explains how magnetic forces can exert rotational effects on current loops or dipoles.

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