Question

A hairpin configuration is formed of two semi-infinite straight wires that are \(2.00 \mathrm{~cm}\) apart and joined by a semicircular piece of wire (whose radius must be \(1.00 \mathrm{~cm}\) and whose center is at the origin of \(x y z\) -coordinates ). The top wire lies along the line \(y=1.00 \mathrm{~cm},\) and the bottom wire lies along the line \(y=-1.00 \mathrm{~cm} ;\) these two wires are in the left side \((x<0)\) of the \(x y\) -plane. The current in the hairpin is \(3.00 \mathrm{~A},\) and it is directed toward the right in the top wire, clockwise around the semicircle, and to the left in the bottom wire. Find the magnetic field at the origin of the coordinate system.

Short answer

Question: Calculate the net magnetic field at the origin (\(0,0,0\)) due to a hairpin configuration of wires carrying a current of \(3.00\) A. Answer: The net magnetic field at the origin is approximately \(\vec{B} = 1.50 \times 10^{-6}(ln(2) + \frac{1}{2})\ \mathrm{T}\ \hat{k}\).

Step-by-step solution

Determine the Biot-Savart law for the magnetic field due to a current element

Recall that the magnetic field due to a small current element \(I \ d\vec{l}\) at a position \(\vec{r}\) is given by the Biot-Savart law: \(d\vec{B}=\frac{\mu_0}{4\pi} \cdot \frac{I \ (d\vec{l}\times\hat{r})}{r^2}\) Where \(I\) is the current, \(\mu_0\) is the permeability of free space, and \(\hat{r}\) is the unit direction vector pointing from the current element to the point \(\vec{r}\).

Determine the magnetic field due to the top wire

Let's first solve for the magnetic field at the origin \((0,0,0)\) due to the top wire. Since the wire is located along the line \(y = 1.00\) cm, we can define the current element \(d\vec{l}\) as \(d\vec{l} = dx \ \hat{i}\). The position vector of the point \((0,0,0)\) relative to the point on the wire \((x,1,0)\) is \(\vec{r} = -x \ \hat{i} - 1.00 \ \hat{j}\), which has a magnitude of \(r=\sqrt{x^2+1^2}\). Thus, the unit vector in the direction of \(\vec{r}\) is \(\hat{r} = \frac{-x\hat{i} - 1.00\hat{j}}{\sqrt{x^2+1^2}}\). Using the Biot-Savart law, we compute the magnetic field contribution at the origin due to this element of the wire: \(d\vec{B}_1 = \frac{\mu_0}{4\pi} \cdot \frac{3.00 \ (dx \ \hat{i} \times (-x \ \hat{i} - 1.00 \ \hat{j})/\sqrt{x^2+1^2})}{(x^2+1^2)}\) Since \(\hat{i} \times \hat{i}=0\), we are left with: \(d\vec{B}_1=- \frac{(-\mu_0)(3.00)(1)}{4\pi(x^2+1^2)}dx \ \hat{k}\) Now, we need to integrate \(d\vec{B}_1\) from the left side (\(x \rightarrow -\infty\)) to the semicircle point \((x=0)\) to get the magnetic field at the origin due to the top wire. \(\vec{B}_1 = \int_{-\infty}^{0}- \frac{(-\mu_0)(3.00)(1)}{4\pi(x^2+1^2)}dx \ \hat{k}\)

Determine the magnetic field due to the bottom wire

Now, let's find the magnetic field at the origin due to the bottom wire, which is located along the line \(y=-1.00\) cm. Similarly to the case of the top wire: \(d\vec{B}_2 = -\frac{\mu_0I}{4\pi(x^2+1^2)}dx \ (-\hat{k})\) Integrating from the left side (\(x \rightarrow -\infty\)) to the semicircle point \((x=0)\): \(\vec{B}_2 = \int_{-\infty}^{0}- \frac{(-\mu_0)(3.00)(1)}{4\pi(x^2+1^2)}dx(-\hat{k})\)

Determine the magnetic field due to the semicircular wire

Lastly, let's compute the magnetic field at the origin due to the semicircle. At an angle \(\theta\) on the semicircle, we can define the current element \(d\vec{l} = rd\theta(-\sin\theta \hat{i} + \cos\theta \hat{j})\). Here, \(r=1.00\). We can integrate the contributions to the magnetic field \(d\vec{B}\) (arising from the Biot-Savart law) over the angle \(\theta\) from \(0\) to \(\pi\) to obtain \(\vec{B}_3\): \(\vec{B}_3 = \frac{1\cdot(-1)^2 \mu_0 I}{4} \int_{0}^{\pi} (\frac{1}{1+1.00\cdot \sin\theta})d\theta \ \hat{k}\)

Determine the net magnetic field at the origin

Now, we can find the total magnetic field at the origin by adding up the contributions from each piece of wire: \(\vec{B}=\vec{B}_1+\vec{B}_2+\vec{B}_3\) After calculation (with the help of a table or integral calculator), \(\vec{B}=\frac{\mu_0I}{2\pi}(ln(2)+\frac{1}{2})\hat{k}\) Substituting the given values and the permeability of free space \(\mu_0=4\pi \times 10^{-7} \ \mathrm{T\cdot m/A}\), we find the magnetic field at the origin: \(\vec{B}=(4\pi \times 10^{-7} \ \mathrm{T\cdot m/A})(3.00 \ \mathrm{A})(\frac{1}{2\pi})(ln(2) + \frac{1}{2})\hat{k} = 1.50 \times 10^{-6}(ln(2) + \frac{1}{2})\ \mathrm{T}\ \hat{k}\)

Key concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental equation in electromagnetism that describes how a current element can create a magnetic field in the surrounding area. It relates a magnetic field, \(d\vec{B}\), to a current element, \(I \(d\vec{l})\), its direction, and the position vector, \(\vec{r}\), from the current element to the point where the magnetic field is being calculated.

In essence, the law says that the magnetic field created by a current is directly proportional to the magnitude of the current and inversely proportional to the square of the distance from the current. The direction of the magnetic field is perpendicular to both the current and the position vector. The Biot-Savart Law can be expressed mathematically as:
\[d\vec{B} = \frac{\mu_0}{4\pi} \cdot \frac{I \ (d\vec{l}\times\hat{r})}{r^2}\]
where \(\mu_0\) is the permeability of free space, \(I\) is the current, and \(\hat{r}\) is the unit vector pointing from the current element to the point of interest. This relationship is crucial for calculating the magnetic field produced by different configurations of current-carrying wires.

Current-Carrying Wires

When current flows through a wire, it generates a magnetic field around it. This field can be described by the Biot-Savart Law, as mentioned earlier. The magnetic field lines form concentric circles around the wire, with the strength of the field depending on the distance from the wire and the amount of current running through it.

The magnetic field's direction can be determined by the right-hand rule: if you point the thumb of your right hand in the direction of the current, your fingers will curl in the direction of the magnetic field lines. For straight, current-carrying wires, the magnetic field at a point is determined by integrating the contributions of infinitesimal current elements along the length of the wire.

Magnetic Field Due to a Current Element

In the case of a current element, which can be thought of as an infinitesimally small segment of a wire carrying current, the Biot-Savart Law enables us to calculate the infinitesimal contribution to the magnetic field, known as \(d\vec{B}\).

For example, if we have a straight wire carrying current, each segment \(d\vec{l}\) contributes an infinitesimal part to the overall magnetic field. By integrating these infinitesimal contributions along the length of the wire, we can determine the total magnetic field. This technique is especially useful when dealing with complex geometries of wires, such as loops or coils, where the contribution of each segment must be calculated with respect to its orientation and position relative to the point of interest.

Superposition Principle in Magnetism

The superposition principle is a critical concept in physics that applies to magnetic fields among other phenomena. It states that the total magnetic field created by multiple sources is the vector sum of the fields produced by each source independently.

This principle simplifies complex problems, like the one in our exercise involving a hairpin setup of wires. We first determine the magnetic field contribution from each segment of the configuration - the top wire, the bottom wire, and the semicircular piece - and then sum up these contributions to find the total magnetic field at a point. Mathematically, this is expressed as:
\[\vec{B} = \vec{B}_1 + \vec{B}_2 + \vec{B}_3\]
where \(\vec{B}_1\), \(\vec{B}_2\), and \(\vec{B}_3\) are the magnetic fields due to each individual segment of the current-carrying configuration. By applying the superposition principle, we can handle more complicated arrangements of currents and accurately predict the resultant magnetic field.