91Ó°ÊÓ

Calculating the magnetic field due to a moving point charge is a classic exercise in electromagnetism. One of the fundamental tools we use for this is the Biot-Savart Law. This law provides a way to calculate the magnetic field (\( \vec{B} \)) that a moving charge generates at any specific point in space.
The main formula looks like this: \[\vec{B} = \frac{\mu_0}{4\pi} \cdot \frac{q(\vec{v} \times \vec{r})}{r^3}\]Here's a quick breakdown of what these symbols mean:

• \( \mu_0 = 4\pi \times 10^{-7} \,\text{T}\cdot\text{m/A} \), the permeability of free space.
• \( q \) represents the charge in coulombs.
• \( \vec{v} \) is the velocity vector of the charge.
• \( \vec{r} \) is the position vector from the charge to the point where we want to calculate the field.
• \( r \) is the magnitude of the vector \( \vec{r} \).

To solve these types of problems, we frequently need to find the cross product of the velocity and position vectors and then plug everything into the formula. The calculations might seem complex, but they are manageable if you handle them step by step.

A charge in motion behaves differently than a stationary charge. Stationary charges produce electric fields, while moving charges create both electric and magnetic fields. For our exercise, we have a charge of \(+6.00-\mu \mathrm{C} \) moving at \(8.00 \times 10^{6} \,\text{m/s}\).
Motion vector, \( \vec{v} \), indicates the path and speed of the movement. For this problem, the charge is moving in the \(+y\)-direction.

• When a charge moves through space, it interacts with the magnetic field of the point of interest at different angles.
• This is why vectors are vital—they help us determine the orientation and magnitude changes of the field.

The direction of the magnetic field can change with the position vector, while the same charge moves, leading to different results at every point.

The cross product is an essential mathematical operation when dealing with vectors, particularly in calculating magnetic fields with moving charges. It's used to find a vector that is perpendicular to two other vectors.
For our exercise:

• Velocity vector \( \vec{v} = \langle 0, 8.00 \times 10^6, 0 \rangle \).
• Position vector \( \vec{r} \) varies depending on the point of interest, like \( \langle 0.5, 0, 0 \rangle \) for point (a).

The cross product of \( \vec{v} \times \vec{r} \) gives us another vector pointing out of these plane, which indicates the orientation of the magnetic field:\[\vec{v} \times \vec{r} = \langle x, y, z \rangle\]The resulting vector's components impact the magnetic field vector through the Biot-Savart Law, affecting both direction and magnitude.

Position vectors are crucial for pinpointing where in space we want to calculate effects like magnetic fields. They extend from the source of charge to the specific location where the field is determined.
In our solution, the position vector \( \vec{r} \) changes with each part, a through d, highlighting different points in space relative to the charge:

• Point (a) has \( \vec{r} = \langle 0.5, 0, 0 \rangle \)
• Point (b) has \( \vec{r} = \langle 0, -0.5, 0 \rangle \)
• Point (c) has \( \vec{r} = \langle 0, 0, 0.5 \rangle \)
• Point (d) has \( \vec{r} = \langle 0, -0.5, 0.5 \rangle \)

To find the magnitude of these vectors, we compute:\[ r = \sqrt{x^2 + y^2 + z^2} \]This magnitude helps in finding out the contribution of each position to the magnetic field using the Biot-Savart law. Shifts in any coordinate dramatically affect both the calculation and the orientation of \( \vec{B} \).