Question

An element \(\mathrm{d} \ell^{-}=\mathrm{dx} \uparrow\) (where \(\mathrm{dx}=1 \mathrm{~cm}\) ) is placed at the origin and carries a large current \(\mathrm{I}=10 \mathrm{Amp}\). What is the mag. field on the Y-axis at a distance of \(0.5\) meter ? (a) \(2 \times 10^{-8} \mathrm{k} \wedge \mathrm{T}\) (b) \(4 \times 10^{8} \mathrm{k} \wedge \mathrm{T}\) (c) \(-2 \times 10^{-8} \mathrm{k} \wedge \mathrm{T}\) (d) \(-4 \times 10^{-8} \mathrm{k} \wedge \mathrm{T}\)

Short answer

The magnetic field on the Y-axis at a distance of 0.5 meters is \(-2 \times 10^{-8} \mathrm{k} \wedge \mathrm{T}\).

Step-by-step solution

Biot-Savart Law

The Biot-Savart Law states that the magnetic field (dB) created by a small element \(d\ell\), carrying a current I, can be found by the following formula: \[dB= \frac{\mu_0}{4\pi}\cdot \frac{I \cdot d\ell \times \hat{r}}{r^2}\] where \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7} Tm/A\)), \(d\ell\) is the length of the element, \(\hat{r}\) is the unit vector pointing from the element to the point of interest, and r is the distance from the element to the point of interest.

Set up the Biot-Savart Integral

Since we have a small element \(d\ell^-\) at the origin, the integral for finding the magnetic field at a point on the Y-axis can be set up as: \[B = \int dB = \int \frac{\mu_0}{4\pi}\cdot \frac{I \cdot d\ell \times \hat{r}}{r^2}\]

Calculate the vector cross product

Since \(d\ell\)= dx⇑ and the point P is on the positive Y-axis, the unit vector \(\hat{r}\) is in the positive Y direction as well. Evaluating the cross product dℓ × \(\hat{r}\) gives: \(d\ell \times \hat{r}\) = (dx⇑) × (0,1,0) = (0,0,-dx)

Determine the distance

Since the point P is on the positive Y-axis at a distance of 0.5 meters (50 cm) from the origin: \(r = 50\ cm\)

Evaluate the Biot-Savart integral

Now, we plug the values of the cross product and distance back into the integral: \[B = \int dB = \int \frac{\mu_0}{4\pi}\cdot \frac{I \cdot (0,0,-dx)}{50^2}\] \[B_z = -\int_{0}^{1} \frac{\mu_0 \cdot I \cdot dx}{4\pi \cdot 50^2}\] Finishing the integral, we get: \[B_z = - \frac{\mu_0 I}{4\pi \cdot 50^2}\] Now, plug in the values for \(\mu_0\) and I: \[B_z = - \frac{4\pi \times 10^{-7} \: Tm/A \cdot 10 \: A}{4\pi \cdot 50^2}\]

Solve for the magnetic field

Simplifying and solving for the magnetic field, we get: \[B_z = -2 \times 10^{-8} \: T\] Therefore, the correct answer is (c) \(-2 \times 10^{-8} \mathrm{k} \wedge \mathrm{T}\).

Key concepts

Magnetic Field Calculation

Understanding the magnetic field calculation is essential when dealing with magnetic forces generated by electric currents. The Biot-Savart Law is a key tool here. It provides a mathematical model for calculating the magnetic field (\(dB\)) generated by a small segment of current-carrying conductor (\(d\ell\)). The general formula for this calculation is: \[dB = \frac{\mu_0}{4\pi} \cdot \frac{I \cdot d\ell \times \hat{r}}{r^2}\] In this equation, - \(\mu_0\) is the permeability of free space, - \(I\) represents the current, - \(d\ell\) signifies a small length of conductor, - \(\hat{r}\) is the unit vector pointing from the current element to the point where the field is calculated, and - \(r\) is the distance from the current element to the point of interest.To comprehend the magnetic field calculation process, consider each component's role in the Biot-Savart Law. The interplay between these variables allows us to compute the magnetic influence that originates from electric currents.

Current Element

The concept of the current element \(d\ell\) is pivotal when applying the Biot-Savart Law. Essentially, this element is a small segment of a wire carrying current. Understanding it is crucial, especially when calculating magnetic fields created by more extensive setups.In the provided exercise, \(d\ell\) is represented as a formulaic segment aligned along the x-axis with a specific length, like \(\mathrm{dx} = 1\ cm\). This scenario involves a small piece of conductor at the origin carrying a sizeable current \(I = 10\ Amp\). Despite being a miniature segment, this element is central in understanding how a magnetic field forms and behaves around it.In practical applications: - The direction of \(d\ell\) follows the direction of the current flow.- Each current element contributes to the overall magnetic field.- Summing, or integrating, these contributions yields the total magnetic effect at a given point.Thus, even tiny segments must be examined meticulously to build an accurate picture of the magnetic field generated by a current.

Cross Product in Physics

The cross product, appearing in the Biot-Savart Law, is a fundamental concept in physics that quantifies the effect of two vectors in different planes. For magnetic fields, we specifically look at how the current element vector \(d\ell\) interacts with the radial unit vector \(\hat{r}\) through cross product calculations.This physics operation follows these rules: - The ___cross product produces a vector perpendicular to the plane formed by the two input vectors___.- Its magnitude depends on the product of the magnitudes of two vectors and the sine of the angle between them.- The direction follows the right-hand rule, aiding correct vector orientation.In the given example, \(d\ell\) is vertical, while the unit vector \(\hat{r}\) is horizontal, resulting in:\[d\ell \times \hat{r} = (dx \uparrow) \times (0, 1, 0) = (0, 0, -dx)\]Here, the resultant vector is along the negative z-axis, consistent with the cross product rules. Such calculations help us determine the magnetic field direction and magnitude influenced by the current.

Permeability of Free Space

The permeability of free space (\(\mu_0\)) plays an important role in magnetic field calculations, serving as a proportional constant in the Biot-Savart Law. This constant characterizes the extent to which a magnetic field can penetrate and permeate through a vacuum.The standard value is: \[\mu_0 = 4\pi \times 10^{-7} \ Tm/A\]This value is crucial for equations predicting magnetic fields due to currents.Key aspects include: - This constant bridges the relationship between electricity and magnetism, as described by Maxwell's equations.- In calculations, \(\mu_0\) scales the effect of a given current size on the magnetic field strength.- Its consistency aids in establishing universal calculations for magnetic interactions.In the exercise provided, \(\mu_0\) directly influences how we predict the magnetic field at a point 0.5 meters from a current element at the origin. Recognizing this constant's role enhances comprehension of magnetic phenomena in various physical contexts.