Question

Two infinitely long straight wires lie in the \(x y\) -plane along the lines \(x=\pm R\). The wire located at \(x=+R\) carries a constant current \(I_{1}\) and the wire located at \(x=-R\) carries a constant current \(I_{2} .\) A circular loop of radius \(R\) is suspended with its centre at \((0,0, \sqrt{3} R)\) and in a plane parallel to the \(x y\) -plane. This loop carries a constant current \(I\) in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the \(+\hat{\jmath}\) direction. Which of the following statements regarding the magnetic field \(\vec{B}\) is (are) true? (A) If \(I_{1}=I_{2}\), then \(\vec{B}\) cannot be equal to zero at the origin \((0,0,0)\) (B) If \(I_{1}>0\) and \(I_{2}<0\), then \(\vec{B}\) can be equal to zero at the origin \((0,0,0)\) (C) If \(I_{1}<0\) and \(I_{2}>0\), then \(\vec{B}\) can be equal to zero at the origin \((0,0,0)\) (D) If \(I_{1}=I_{2}\), then the \(z\) -component of the magnetic field at the centre of the loop is \(\left(-\frac{\mu_{0} I}{2 R}\right)\)

Short answer

(A) False, because the loop's current still contributes to the magnetic field at the origin. (B) and (C) are False, because the net magnetic field due to the loop cannot be canceled by the fields of wires carrying currents in opposite directions. (D) True, because the loop contributes to a negative \(z\)-component magnetic field at its center.

Step-by-step solution

Assess the Magnetic Field from Each Wire at the Origin

The magnetic field contribution from a long straight wire at a point perpendicular to the wire at a distance R is given by the Biot-Savart Law, which states that the magnitude of the magnetic field at this point is \(B = \frac{\mu_0 I}{2\pi R}\). The direction of the magnetic field is given by the right-hand rule. The magnetic field from wire 1 (at \(x=+R\)) will be into the page, and the magnetic field from wire 2 (at \(x=-R\)) will be out of the page when both carry positive currents in the \(+\hat{\jmath}\) direction.

Determine the Net Magnetic Field at the Origin for Condition (A)

If \(I_1=I_2\), the magnetic field at the origin due to wire 1 will be equal in magnitude but opposite in direction to the magnetic field due to wire 2, thus leading to a cancellation in the \(x y\)-plane. However, there is no consideration of the magnetic field contributions in the \(z\)-direction. Since the loop has a current flowing in the clockwise direction, its contribution to the magnetic field at the origin is not zero. Therefore, the total magnetic field at the origin cannot be equal to zero.

Evaluate the Conditions for the Net Magnetic Field to be Zero at the Origin for Conditions (B) and (C)

For the net magnetic field to be zero at the origin, the magnetic fields due to both wires and the loop must cancel out. When \(I_1>0\) and \(I_2<0\), both wires contribute magnetic fields in opposite directions along the \(xy\)-plane, and the loop's contribution is along the \(z\)-direction. If the magnitudes of the currents are such that the resulting magnetic fields from the wires cancel each other in the \(xy\)-plane, then the remaining magnetic field from the loop in the \(z\)-direction will still be non-zero, preventing the total magnetic field from being zero. Conversely, when \(I_1<0\) and \(I_2>0\), a similar analysis applies. The magnetic fields from the wires will still be in opposite directions, and if they cancel each other, the loop's field will remain, again preventing the total magnetic field from being zero at the origin.

Calculate the Z-Component of the Magnetic Field at the Center of the Loop for Condition (D)

The center of the loop is at a distance of \(\sqrt{3}R\) from the \(xy\)-plane. The contribution to the magnetic field in the \(z\)-direction from currents in the wires at \(x=\pm R\) is zero because they are symmetrically placed with respect to the origin along the \(x\)-axis, hence they only contribute to the field in the \(xy\)-plane if \(I_1=I_2\). The current in the loop itself creates a magnetic field in the \(z\)-direction at its center, which can be calculated using Ampere's Law. For a current carrying loop, the magnetic field at the center is given by \( B = \frac{\mu_0 I}{2 R} \), pointing in the negative \(z\)-direction (due to the clockwise current when viewed from above). Therefore, the \(z\)-component of the magnetic field at the center of the loop is \( -\frac{\mu_0 I}{2 R} \) if \(I_1=I_2\).

Key concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle in electromagnetism that describes how current-carrying wires generate magnetic fields. According to this law, the magnetic field \(\vec{B}\) at a point in space due to an infinitesimal element of current is directly proportional to the current \(I\), inversely proportional to the square of the distance from the element to the point, and depends on the angle between the element and the line joining it to the point.

For a long straight wire, the law simplifies to determine the magnetic field at a point that is perpendicular to the wire at a distance \(R\). The magnitude of this field is given by \(B = \frac{\mu_0 I}{2\pi R}\), where \(\mu_0\) is the permeability of free space. The direction of \(\vec{B}\) is perpendicular to both the current direction and the line connecting the wire to the point, following the right-hand rule.

The understanding of the Biot-Savart Law is crucial when assessing complex magnetic field scenarios, such as the one described in our exercise, where multiple currents contribute to the total magnetic field at a given point.

Ampere's Law

Ampere's Law is another vital principle used in electromagnetism, particularly when dealing with symmetrical current distributions. It states that the magnetic field \(\vec{B}\) along a closed loop is equal to \(\mu_0\) times the total current \(I\) enclosed by the loop. Mathematically, Ampere's Law is represented as \(\oint \vec{B} \cdot d\vec{l} = \mu_0 I\), where the integral is performed over the closed loop.

For simple cases such as a circular loop of wire, Ampere's Law allows us to easily calculate the magnetic field at the center of the loop. The application of Ampere's Law to our exercise is evident in the estimation of the \(z\)-component of the magnetic field at the center of the loop, which manifests because of the current flowing in the circular loop.

Right-Hand Rule

The right-hand rule is a mnemonic for understanding orientation in vector fields, such as magnetic fields created by currents. To apply the rule for current-carrying wires, you point your thumb in the direction of the current, and the curl of your fingers shows the direction in which the magnetic field \(\vec{B}\) circulates around the wire.

Using the right-hand rule enables us to determine the direction of magnetic fields in our exercise. For instance, it helps ascertain that the magnetic field from wire 1 at \(x=+R\) will be directed into the page and the magnetic field from wire 2 at \(x=-R\) will be directed out of the page when they both carry positive current in the \(+\hat{\jmath}\) direction.

Magnetic Field Cancellation

Magnetic field cancellation occurs when the magnetic fields from multiple sources effectively negate each other, resulting in a net field of zero at a particular point. This phenomenon is pivotal in understanding scenarios involving parallel current-carrying wires with currents in opposite directions.

In the context of our exercise, we consider the magnetic fields at the origin due to two long straight wires. If these wires carry currents of equal magnitude but opposite directions, their contributions to the magnetic field in the \(xy\)-plane would cancel. However, this does not account for other sources, like the circular loop carrying current above the origin. Thus, for complete field cancellation at a point, all individual magnetic field components, including those in the \(z\)-direction, must cancel out.

Current Distribution

The distribution of electric current in conductors affects the magnetic field in the surrounding region. Symmetry in current distribution plays a crucial role in determining the resulting field. In our exercise, we examine currents in two long straight wires lying along the lines \(x=\pm R\) and a current in a circular loop centered above the origin.

The specific arrangement and orientation of the wires and the loop influence the magnetic field at different points in space. The direction of the current in each conductor (whether it is positive or negative) will affect the direction of the magnetic field they produce. Understanding current distribution helps in analyzing the magnetic field within a region and is essential in calculating the net magnetic field at any point given a configuration of current-carrying conductors, as shown in the solutions for conditions (B) and (C) of the exercise.