Question

A current of 6 A flows from $M(2,0,5)$ to $N(5,0,5)$ in a straight, solid conductor in free space. An infinite current filament lies along the $z$ axis and carries $50\mathrm{A}$ in the ${\mathbf{a}}_z$ direction. Compute the vector torque on the wire segment using an origin at: $(a)(0,0,5);(b)(0,0,0);(c)(3,0,0)$.

Short answer

Question: Calculate the vector torque on a straight, solid conductor with a current of 6 A flowing from point M(2, 0, 5) to point N(5, 0, 5) due to an infinite current filament along the z-axis carrying 50 A in the az direction using different origins: (a) (0, 0, 5); (b) (0, 0, 0); and (c) (3, 0, 0). Answer: To calculate the vector torque, perform the following steps: 1. Calculate the magnetic field due to the infinite current filament at each origin using Biot-Savart law. 2. Calculate the positions of point M and N relative to each origin. 3. Calculate the magnetic field at points M and N for each origin. 4. Calculate the magnetic force on the wire segment for each origin. 5. Calculate the torque for each origin using the respective magnetic forces and positions. By substituting the obtained values in step 5, we can find the torque for each origin.

Step-by-step solution

Calculate Magnetic Field due to the Infinite Current Filament at the Origin

We can use the Biot-Savart law to calculate the magnetic field produced by the infinite current filament. The magnetic field $B_z$ at a radial distance $r$ from the z-axis is: $$B_z=\frac{{\mu }_0{I}_{filament}}{2\pi r}$$ Where $B_z$ is the magnetic field due to the filament, ${\mu }_0$ is the permeability of free space ($(4\pi \ast {10}^{-7})\frac{Tm}{A}$), $I_{filament}$ is the current in the filament (50 A), and $r$ is the radial distance from the filament to the point at which we want to calculate the magnetic field. We will calculate the magnetic field at different origins.

Calculate the Positions of Point M and N Relative to Each Origin

To compute the torque, we need the positions of points M and N relative to the different origins. Let the positions of points M and N relative to each origin be $r_M$ and $r_N$ respectively. We can find these positions by subtracting the corresponding coordinates. (a) Origin (0,0,5) $$r_M=(2,0,0)$$ $$r_N=(5,0,0)$$ (b) Origin (0,0,0) $$r_M=(2,0,5)$$ $$r_N=(5,0,5)$$ (c) Origin (3,0,0) $$r_M=(-1,0,5)$$ $$r_N=(2,0,5)$$

Calculate the Magnetic Field at Points M and N for Each Origin

Next, we need to calculate the magnetic field at points M and N for each origin. Using the magnetic field formula and the radial distances found in step 2: $$B_z=\frac{{\mu }_0{I}_{filament}}{2\pi r}$$ For origin (a) (0,0,5) : $$B_{z_M}=\frac{{\mu }_{0}50}{2\pi 2}$$ $$B_{z_N}=\frac{{\mu }_{0}50}{2\pi 5}$$ For origin (b) (0,0,0) : $$B_{z_M}=\frac{{\mu }_{0}50}{2\pi \sqrt{2^2+5^2}}$$ $$B_{z_N}=\frac{{\mu }_{0}50}{2\pi \sqrt{5^2+5^2}}$$ For origin (c) (3,0,0) : $$B_{z_M}=\frac{{\mu }_{0}50}{2\pi \sqrt{(-1{)}^2+5^2}}$$ $$B_{z_N}=\frac{{\mu }_{0}50}{2\pi \sqrt{2^2+5^2}}$$

Calculate the Magnetic Force on the Wire Segment

We can calculate the magnetic force on the wire segment using the following formula, where $F_M$ and $F_N$ are forces on points M and N, $I$ is the current in the wire, $dl=r_N-r_M$, and $B$ is the magnetic field at each point. $$F_M=Idl\times B_M$$ $$F_N=Idl\times B_N$$ We need to calculate $F_M$ and $F_N$ for each origin.

Calculate the Torque for Each Origin

Finally, we need to find the torque on the wire segment for each origin using the following formulas: $$Torque=r_M\times F_M+r_N\times F_N$$ Substitute the values obtained in previous steps and calculate the torque for each origin.

Key concepts

Understanding the Biot-Savart Law

When discussing magnetic fields and their sources, we cannot overlook the significance of the Biot-Savart Law. This fundamental physical principle helps us to calculate the magnetic field generated by a steady current. Imagine a tiny segment of current-carrying wire; according to the Biot-Savart Law, this segment produces a magnetic field that decreases with distance and depends on the current's direction and magnitude.

Mathematically, the Biot-Savart Law is expressed as:
$$B=\frac{{\mu }_0}{4\pi }\int \frac{Id\overrightarrow{s}\times \hat{r}}{r^2}$$
Here, $B$ is the magnetic field, ${\mu }_0$ represents the permeability of free space, $I$ is the current, $d\overrightarrow{s}$ is the infinitesimal element of the wire carrying the current, $\hat{r}$ is the unit vector from the wire element pointing towards the observation point, and $r$ is the distance from the wire element to the observation point.

Utilizing the Biot-Savart Law requires us to consider both the magnitude and direction of the current as well as the shape of the current-carrying conductor, since the resulting field is a vector that must be integrated over the path of the current. For more complex geometries, this might involve some advanced calculus, but the underlying principle remains the same: the combination of current flow and distance influences the magnetic field’s strength and orientation.

Magnetic Field Calculation around a Wire

Calculating the magnetic field around a wire involves understanding how the magnetic field is distributed in space due to the current passing through the wire. Especially when dealing with an infinite current filament, as in our exercise, the symmetry of the problem simplifies the calculation.

For an infinite straight wire, the magnetic field follows a circular pattern surrounding the wire and diminishes with distance as per the modified Biot-Savart Law:$$B=\frac{{\mu }_{0}I}{2\pi r}$$
This formula tells us that the magnetic field $B$ is directly proportional to the current $I$ and inversely proportional to the distance $r$ from the wire. Furthermore, the direction of the field follows the right-hand rule, which states that if you wrap your right hand around the wire with the thumb pointing in the direction of the current, your fingers will curl in the direction of the magnetic field lines.

When solving problems like the one in our textbook, we use this formula to find the magnetic field at specific points in space, taking into account their distance from the current filament. Let's not forget that for points along the axis of the filament, the magnetic field would be null since the distance $r$ is zero, thus leading to a zero division in the equation—which is not physically meaningful.

Vector Torque Computation on a Current-Carrying Wire

The concept of torque is central in understanding how forces can cause objects to rotate. When it comes to a magnetic context, like our textbook problem, torque can be exerted on a current-carrying wire segment in a magnetic field. This torque is the measure of the wire's tendency to rotate about an axis due to the magnetic forces.

The computation of magnetic torque involves vector cross products, represented by the equation:$$\tau =\overrightarrow{r}\times \overrightarrow{F}$$
In this equation, $\tau$ denotes the torque, $\overrightarrow{r}$ is the position vector from the pivot point to the point where the force is applied, and $\overrightarrow{F}$ is the magnetic force acting on the wire. The cross product means that the torque is maximized when the position vector and force vector are perpendicular to each other and is zero when they are parallel.

For the wire segment in our exercise, we find that the torque not only depends on the magnitude of the current and magnetic field but also on the position of the segment in relation to different chosen origins. After calculating the forces at points M and N due to the magnetic field, we use the position vectors relative to each origin point and the right-hand rule to determine the direction of the torque. Applying the cross-product, we establish the exerted torque, revealing potential rotational effects on the wire segment.

Understanding these fundamental concepts not only helps solve textbook equations but also provides insights into real-world applications, such as electric motors where torque plays a critical role in their operation.