Question

A single-turn coil \(C\) of arbitrary shape is placed in a magnetic field \(\mathbf{B}\) and carries a current \(I\). Show that the couple acting upon the coil can be written as $$ \mathbf{M}=I \int_{C}(\mathbf{B} \cdot \mathbf{r}) d \mathbf{r}-I \int_{C} \mathbf{B}(\mathbf{r} \cdot d \mathbf{r}) $$ For a planar rectangular coil of sides \(2 a\) and \(2 b\) placed with its plane vertical and at an angle \(\phi\) to a uniform horizontal field \(\mathbf{B}\), show that \(\mathbf{M}\) is, as expected, \(4 a b B I \cos \phi \mathbf{k}\).

Short answer

Couple on the coil: \(\text{M} = I \int_{C}(\mathbf{B} \cdot \mathbf{r}) d \mathbf{r}- I\int_{C} \mathbf{B} ( \mathbf{r}\cdot d \mathbf{r}) \). For the rectangular coil: \(\text{M} = 4abBI\text{cos} \phi \textbf{k} \).

Step-by-step solution

- Understanding the Couple Acting on the Coil

The couple (or torque) on a current-carrying coil in a magnetic field can be given by \( \textbf{M} = I \textbf{A} \times \textbf{B} \), where \( \textbf{A} \) is the area vector of the coil. However, for an arbitrary shape, we need a more general form.

- Starting with the Magnetic Moment

The magnetic moment \( \textbf{m} \) of the coil is \( I \textbf{A} \), where \( I \) is the current and \( \textbf{A} \) is the area vector (giving both the magnitude and direction of the area).

- Applying the Lorentz Force

The force on an element of the coil \( d\textbf{r} \) due to the magnetic field \( \textbf{B} \) is given by \( d\textbf{F} = I d\textbf{r} \times \textbf{B} \).

- Computing the Total Couple

The torque \( \textbf{M} \) is the integral of \( \textbf{r} \times d\textbf{F} \). Thus, \( \textbf{M} = I \times \text{{int}}_{C} \textbf{r} \times (d\textbf{r} \times \textbf{B}) \). Expanding this triple product, we get \( \textbf{M} = I \text{{int}}_{C} (\textbf{B} \times (\textbf{r} \times d\textbf{r})).\)

- Using Vector Triple Product Properties

Using \((\textbf{A} \times (\textbf{B} \times \textbf{C})) = (\textbf{A} \bullet \textbf{C}) \textbf{B} - (\textbf{A} \bullet \textbf{B}) \textbf{C}\), we get \( I \text{{int}}_{C} (\textbf{r} \bullet \textbf{B}) d\textbf{r}) - I \text{{int}}_{C} \textbf{B} (\textbf{r} \bullet d \textbf{r}), \) leading to the final form.

- Applying to Rectangular Coil in Uniform Field

For a planar rectangular coil of sides \(2a\) and \(2b\), with the plane at an angle \( \theta \) to the field \( \textbf{B} \), the area vector magnitude is \(A = 4ab\) and it makes an angle \( \theta \) with \( \textbf{B} \). Therefore, the magnitude of the couple is \( |\textbf{M}| = IA B \text{{cos}} \theta \). As the plane is vertical, the direction will be \( \textbf{k} \), making \( \textbf{M} = 4ab IB \text{{cos}} \theta \textbf{k}. \)

Key concepts

Magnetic Field

A magnetic field, denoted as \(\textbf{B}\), is a vector field that surrounds magnetic materials and moving electric charges. It exerts a force on other nearby moving charges and magnetic dipoles. In this exercise, the magnetic field influences the current-carrying coil, producing a couple or torque. The direction and magnitude of \(\textbf{B}\) play a vital role in determining the resultant torque on any current-carrying conductor within the field.

• It is measured in Tesla (T).
• Its direction is represented by field lines pointing from the North to the South Pole of a magnet.

Magnetic Moment

The magnetic moment (\(\textbf{m}\)) of a coil is a quantity that represents the strength and orientation of the magnetism of an object. For a coil carrying a current \(I\) and having an area \(A\), the magnetic moment is given by \( \textbf{m} = I \textbf{A} \).

• This vector lies perpendicular to the plane of the coil.
• It indicates how much torque the coil can experience in a given magnetic field.

For the specific problem of a rectangular coil, the area vector \( \textbf{A} \) becomes crucial in calculating the total torque in a magnetic field.

Lorentz Force

The Lorentz force describes the force exerted on a charged particle moving through a magnetic and electric field. In the context of a current-carrying coil in a magnetic field, this force can be described as \( d\textbf{F} = I d\textbf{r} \times \textbf{B} \), where \( d\textbf{r} \) is an infinitesimal element of the coil.

• It is directional and always perpendicular to both the current element and the magnetic field.
• This force is the fundamental reason why a moving charge in a magnetic field experiences torque.

Understanding and applying the Lorentz force law is key to finding the resultant torque on a current-carrying loop in an arbitrary magnetic field.

Vector Triple Product

The vector triple product is a vector identity involving three vectors. When applied to solve magnetic problems, one can use the identity: \( \textbf{A} \times (\textbf{B} \times \textbf{C}) = (\textbf{A} \bullet \textbf{C}) \textbf{B} - (\textbf{A} \bullet \textbf{B}) \textbf{C} \).

• It is essential for expanding and simplifying expressions involving cross products.
• Using this rule helps in solving for the torque or couple acting on the coil.

In the specific solution provided, this identity is employed to obtain the integrals representing the couple on a current-carrying coil in a magnetic field.