Question

Two straight long conductors \(\mathrm{AOB}\) and \(\mathrm{COD}\) are perpendicular to each other and carry currents \(\mathrm{I}_{1}\) and \(\mathrm{I}_{2}\). The magnitude of the mag. field at a point " \(\mathrm{P}^{n}\) at a distance " \(\mathrm{a}^{\prime \prime}\) from the point "O" in a direction perpendicular to the plane \(\mathrm{ABCD}\) is (a) \(\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(I_{1}+I_{2}\right)\) (b) \(\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(I_{1}-I_{2}\right)\) (c) \(\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(\mathrm{I}_{1}^{2}+\mathrm{I}_{2}^{2}\right)^{1 / 2}\) (d) \(\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left[\left(I_{1} I_{2}\right) /\left(I_{1}-I_{2}\right)\right]\)

Short answer

The short answer to the question is: \(B = \frac{\mu_{0}}{2\pi a}\sqrt{I_{1}^{2} + I_{2}^{2}}\) This matches option (c) in the given question.

Step-by-step solution

Apply the Biot-Savart law for conductor AOB

Calculate the magnetic field produced by conductor AOB at point P using the Biot-Savart law, which states that the magnetic field dB at a point due to a small current-carrying segment is given by: \(dB = \frac{\mu_{0}}{4\pi}\frac{I\,d\vec{l} \times \vec{r}}{r^{3}}\) where µ₀ = 4π x 10^(-7) Tm/A is the permeability of free space, I is the current through the conductor, d**l** is the small length of the conductor, **r** is the position vector from the current element to the point where we want to calculate the magnetic field, and r is the magnitude of **r**. For conductor AOB, let 'a' be the distance between point O and point P, and let I₁ be the current flowing through the conductor. The magnetic field at point P in the y-direction is given by: \(B_{y1} = \frac{\mu_{0} I_{1}}{2\pi a}\)

Apply the Biot-Savart law for conductor COD

Calculate the magnetic field produced by conductor COD at point P using the Biot-Savart law. Let I₂ be the current flowing through the conductor. The magnetic field at point P in the x-direction is given by: \(B_{x2} = \frac{\mu_{0} I_{2}}{2\pi a}\)

Calculate the net magnetic field vector at point P

Calculate the magnetic field at point P due to both conductors AOB and COD by finding the vector sum of their respective magnetic fields: \(\vec{B} = B_{y1}\hat{y} + B_{x2}\hat{x}\) Substitute the expressions for \(B_{y1}\) and \(B_{x2}\) from Steps 1 and 2: \(\vec{B} = \left(\frac{\mu_{0} I_{1}}{2\pi a}\right)\hat{y} + \left(\frac{\mu_{0} I_{2}}{2\pi a}\right)\hat{x}\)

Calculate the magnitude of the magnetic field at point P

Calculate the magnitude of the magnetic field at point P using the Pythagorean theorem, where the magnitudes of the x and y components are given from Step 3: \(B = \sqrt{(\vec{B}\cdot\vec{B})} = \sqrt{(\frac{\mu_{0} I_{1}}{2\pi a})^{2} + (\frac{\mu_{0} I_{2}}{2\pi a})^{2}}\) Factor out the common term \(\left(\frac{\mu_{0}}{2\pi a}\right)^{2}\): \(B = \frac{\mu_{0}}{2\pi a}\sqrt{I_{1}^{2} + I_{2}^{2}}\) Comparing the expression we found for the magnetic field magnitude with the given options, we see that it matches option (c): \(B = \left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(\mathrm{I}_{1}^{2}+\mathrm{I}_{2}^{2}\right)^{1 / 2}\) Thus, the correct answer is option (c).

Key concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle in electromagnetism that allows us to calculate the magnetic field generated by a current-carrying conductor. It essentially describes how the magnetic field \(B\) at a point in space is influenced by the current \(I\) flowing through a small segment \(d\vec{l}\) of a conductor. The law is expressed as: \[ dB = \frac{\mu_{0}}{4\pi}\frac{I\,d\vec{l} \times \vec{r}}{r^{3}} \] Here, \(\mu_{0}\) is the permeability of free space, \(d\vec{l}\) is a vector representing a small segment of the conductor, \(\vec{r}\) is the vector from the current element to the point where we are calculating the field, and \(r\) is the magnitude of this vector.

• \(dB\) represents the infinitesimal magnetic field produced by \(d\vec{l}\).
• The cross-product \(d\vec{l} \times \vec{r}\) indicates that the magnetic field is perpendicular to both the current direction and the vector \(\vec{r}\).
• The \(r^{3}\) in the denominator shows that the field decreases rapidly with distance from the conductor.

The Biot-Savart Law is especially useful for determining the magnetic field produced at any point around a conductor, which is critical for understanding how different current configurations influence the magnetic environment.

Current-Carrying Conductor

A current-carrying conductor, such as a straight wire, can create a magnetic field around it. The strength and direction of this magnetic field depend on several factors:

• The magnitude of the current \(I\) flowing through the conductor: Higher currents produce stronger magnetic fields.
• The configuration of the conductor: For example, a loop or a solenoid will have different field distributions compared to a straight wire.
• The distance \(a\) from the conductor: As per the Biot-Savart law, the magnetic field's magnitude decreases with distance \(\frac{1}{a}\).

In our problem, the conductors AOB and COD are perpendicular to each other, with currents \(I_{1}\) and \(I_{2}\). These currents generate magnetic fields \(B_{y1}\) and \(B_{x2}\) at point P, calculated using the formulas derived from the Biot-Savart law:\[ B_{y1} = \frac{\mu_{0} I_{1}}{2\pi a} \] and \[ B_{x2} = \frac{\mu_{0} I_{2}}{2\pi a} \] The presence of two intersecting conductors introduces a scenario wherein their fields need to be added together, necessitating a deeper understanding of vector addition of magnetic fields.

Vector Addition of Magnetic Fields

When multiple currents produce magnetic fields at the same point, like with conductors AOB and COD, we calculate the total magnetic field by vectorially adding the individual fields. Vector addition is necessary because magnetic fields are vector quantities—they have both magnitude and direction. To find the net magnetic field \(\vec{B}\) at point P, one must combine \(B_{y1}\) and \(B_{x2}\) using vector addition: \[\vec{B} = B_{y1}\hat{y} + B_{x2}\hat{x}\] Here, \(B_{y1}\) acts in the \(\hat{y}\) direction and \(B_{x2}\) in the \(\hat{x}\) direction because of their perpendicular arrangement. The magnitude of the resultant field \(B\) can be found using the Pythagorean theorem: \[ B = \sqrt{(B_{y1})^{2} + (B_{x2})^{2}} \] For our exercise, substituting the values obtained from earlier steps gives: \[ B = \frac{\mu_{0}}{2\pi a}\sqrt{I_{1}^{2} + I_{2}^{2}} \] This expression represents the total effect of the two perpendicular fields, showcasing how their components interact through vector addition to form a combined magnetic field at point P.