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 0 (zero).

Step-by-step solution

Set up the integral for the net force

To find the net force on the semicircle, we first need to set up an integral for the force experienced by an infinitesimal length of wire, \(\vec{dF} = i\vec{dl} \times \vec{B}\). The semicircle can be parameterized using polar coordinates as \(\vec{r}(\theta) = R \hat{r}(\theta)\), where the angle \(\theta\) varies from \(-\pi\) to \(\pi\). This parameterization allows us to determine the infinitesimal length vector in polar coordinates, which is given by \(\vec{dl}(\theta) = Rd\theta \hat{\theta}\).

Calculate the cross product

Now we need to calculate the cross product of \(\vec{dl}\) and \(\vec{B}\). Since \(\vec{B}\) points in the positive \(z\)-direction, the cross product simplifies to: \begin{align*} \vec{dF} = i\vec{dl} \times \vec{B} = i(Rd\theta \hat{\theta}) \times (B \hat{z}) = iR B d\theta (\hat{\theta} \times \hat{z})\text{.} \end{align*} To express this cross product in polar coordinates, we recall that \(\hat{\theta} \times \hat{z} = -\hat{r}\); hence, the force experienced by an infinitesimal length of wire is \(\vec{dF} = -iRB d\theta \hat{r}\).

Integrate over the semicircle

Now we integrate the force \(\vec{dF}\) over the entire semicircle: \begin{align*} \vec{F} = \int_{-\pi}^{\pi} \vec{dF} = -i R B\int_{-\pi}^{\pi} d\theta \hat{r}(\theta)\text{.} \end{align*} Since \(\hat{r}(\theta)\) has equal and opposite contributions from each half of the semicircle, the net force on the loop is zero: \begin{align*} \vec{F} = -i R B\int_{-\pi}^{\pi} d\theta \hat{r}(\theta) = \vec{0}\text{.} \end{align*} The net force on the semicircular loop is \(\vec{F}=\vec{0}\).

Key concepts

Magnetic Field

Imagine a magnetic field as a region where magnetic forces can act. This invisible field can be visualized through its effects, such as the alignment of iron filings or the motion of a compass needle.

A simple yet powerful way to represent magnetic fields is through arrows, with each arrow depicting the field's direction and strength at a given point. In our problem, the magnetic field vector, denoted as \(\vec{B}\), points out of the page, following the right-hand rule. The direction is critical, as it influences the outcome when we apply the concept of the cross product to determine the force exerted on a current-carrying wire.

Cross Product in Physics

In physics, the cross product is a binary operation on two vectors in three-dimensional space. It provides a vector perpendicular to both original vectors, with a magnitude equal to the area of the parallelogram that the vectors span.

In the context of our exercise, we use the cross product to calculate the magnetic force on a segment of current-carrying wire. Following the formula \(\vec{F} = i\vec{dl} \times \vec{B}\), the direction of the force is given by the right-hand rule, which in this exercise results in a force perpendicular to the plane of the semicircular loop. Understanding the cross product's properties is vital for predicting the behavior of charged particles and current-carrying conductors in magnetic fields.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is defined by a distance from a reference point and an angle from a reference direction. This system is particularly useful when dealing with curves and shapes like the semicircle in our problem.

In the step-by-step solution, we observe how to parameterize the wire in polar coordinates: \(\vec{r}(\theta) = R \hat{r}(\theta)\), allowing the intuitive integration around the curve. By understanding polar coordinates, students can more easily integrate functions over circular paths and tackle complex geometries in physics and engineering problems.