Question

A semicircular loop of wire of radius \(R\) is in the \(x y\) -plane, centered about the origin. The wire carries a current, \(i\), counterclockwise around the semicircle, from \(x=-R\) to \(x=+R\) on the \(x\) -axis. A magnetic field, \(\vec{B}\), is pointing out of the plane, in the positive \(z\) -direction. Calculate the net force on the semicircular loop.

Short answer

Answer: The net force on the semicircular loop is \(\vec{F_{net}} = 0 \, \hat{i} - 2R \, B \, \hat{j}\).

Step-by-step solution

Express the position vector in terms of the angle

We can express the position vector of a point on the semicircular loop as: \(\vec{r}(\phi) = R\cos \phi \, \hat{i} + R\sin \phi \, \hat{j}\) where \(\phi\) is the angle measured from the negative x-axis, and varies from 0 to \(\pi\).

Calculate the differential length vector

Differentiate the position vector with respect to \(\phi\) to get the differential length vector as a function of angle: \(\frac{d\vec{r}}{d\phi}(\phi) = -R\sin \phi \, \hat{i} + R\cos \phi \, \hat{j}\) Now, we can write the differential length vector as: \(d\vec{L} = \frac{d\vec{r}}{d\phi} d\phi = (-R\sin \phi \, \hat{i} + R\cos \phi \, \hat{j}) d\phi\)

Calculate the magnetic force on a small segment of the wire

Recall that the magnetic force on a small segment of the wire carrying current i is given by: \(d\vec{F} = i \, (d\vec{L} \times \vec{B})\) Since the magnetic field is in the positive z-direction, we have \(\vec{B} = B \, \hat{k}\). Now, we need to find the cross product of dL and B: \(d\vec{L} \times \vec{B} = (-R\sin\phi\, \hat{i} + R\cos\phi\, \hat{j}) \times (B \, \hat{k})\) We can find the magnitude of the force for dyadic cross product representation \(\begin{pmatrix} \hat{i} & \hat{j} & \hat{k} \\ -R\sin \phi & R\cos\phi & 0 \\ 0 & 0 & B \end{pmatrix}\) \(d\vec{F}= - R\cos\phi \, B \hat{i} - R\sin\phi \, B \hat{j} + 0 \hat{k}\)

Integrate the force over the loop

Now, we need to integrate this force over the loop from \(\phi = 0\) to \(\phi = \pi\) to find the net force. The net force along x and y directions can be calculated separately: \(\vec{F_{net}} = \int_{0}^{\pi} d\vec{F} = \int_{0}^{\pi} (-R\cos\phi \, B \, \hat{i} - R\sin\phi \, B \, \hat{j}) \, d\phi\) Integrating along x-direction: \(\vec{F_{x}} = -R\, B \int_{0}^{\pi}\cos\phi \, d\phi = 0\) Integrating along y-direction: \(\vec{F_{y}} = -R \, B \int_{0}^{\pi}\sin\phi \, d\phi = -2R \, B\) Thus, the net force on the semicircular loop is \(\vec{F_{net}} = 0 \, \hat{i} - 2R \, B \, \hat{j}\).

Key concepts

Magnetic Field

A magnetic field is an invisible force field created by magnetic objects, like magnets, or by moving electric charges, such as electric currents. It's usually represented by the symbol \( \vec{B} \) and measured in Teslas (T) in the International System of Units.

Magnetic fields exert forces on other magnetic materials or moving electric charges within the field. This fundamental concept is pivotal in understanding why a current-carrying wire would experience a force when placed in such a field. Additional information includes that magnetic fields can penetrate through all materials, although the field's intensity might decrease depending on the medium.

Lorentz Force

The Lorentz force is the total force experienced by a charged particle moving in an electromagnetic field and is the sum of electric and magnetic forces. When the particle is an electron or a current element \( i \) in a wire, the force is determined by the charge, the velocity of the particle, and the strength of the magnetic and electric fields.

When we only consider the magnetic component, the Lorentz force \( \vec{F} \) on a small segment of current-carrying wire in a magnetic field \( \vec{B} \) is calculated using the formula \( \vec{F} = i (\vec{L} \times \vec{B}) \), where \( i \) is the current and \( \vec{L} \) is the length vector of the wire segment.

Cross Product in Physics

The cross product is a binary operation on two vectors in three-dimensional space, and it results in a vector that is perpendicular to both of the vectors being multiplied. It's denoted \( \vec{A} \times \vec{B} \) and is essential in calculating quantities like torque, angular momentum, and, crucially for our case, the magnetic force.

When calculating the force on a segment of a current-carrying wire \( d\vec{L} \times \vec{B} \) in a magnetic field, we use the cross product to find a vector representing the force that is perpendicular to both the length of the wire segment and the direction of the magnetic field.

Integration in Physics

Integration in physics is used to add up small amounts over a continuous range to find the total quantity, like the net force on a wire in a magnetic field. When we calculate the net magnetic force on a loop, we integrate the differential force \( d\vec{F} \) over the path of the wire.

For our semicircular loop, integration is used to sum the forces at each infinitesimal segment of the wire, considering their directional components. This leads to the final calculation of net force on the wire, which depends on both the shape of the wire and the uniformity of the magnetic field across it.