91Ó°ÊÓ

A magnetic field is an invisible force field created by electric currents and magnets. It is depicted by field lines that run from the north to the south pole. These fields are crucial for understanding how charged particles move in space. In physics, the magnetic field vector is often denoted as \( \vec{B} \). This vector has three components corresponding to its orientation in three-dimensional space:

• \( B_x \): the component along the x-axis
• \( B_y \): the component along the y-axis
• \( B_z \): the component along the z-axis

For our exercise, the magnetic field vector is given by \( \vec{B} = (0.120 \mathrm{T}) \, \hat{i} - (0.220 \mathrm{T}) \, \hat{j} - (0.0900 \mathrm{T}) \, \hat{k} \). Here, each component indicates the strength and direction of the field in that respective axis. Understanding the magnetic field's orientation helps us analyze how the field interacts with moving charges, such as those in a metal rod.

In vector mathematics, the cross product is a method to find a vector that is orthogonal to two given vectors. This is particularly important in physics for calculating induced emf, as it gives us a new vector that helps describe the interaction of velocity and magnetic field vectors. Given two vectors \( \vec{a} \) and \( \vec{b} \), the cross product \( \vec{a} \times \vec{b} \) is computed using the determinant of a matrix involving unit vectors:

• The top row contains the unit vectors \( \hat{i}, \hat{j}, \hat{k} \).
• The second row contains the components of \( \vec{a} \).
• The third row contains the components of \( \vec{b} \).

For the exercise, calculating \( \vec{v} \times \vec{B} \) is accomplished by:\[\vec{v} \times \vec{B} = |\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \ 6.80 & 0 & 0 \ 0.120 & -0.220 & -0.090 \end{array}| \]This determinant expands to yield a new vector, showing how velocity interacts with the magnetic field across different dimensions. This operation results in a vector that is crucial for determining the emf.

The right-hand rule is a mnemonic tool used in vector mathematics, specifically for determining directions in cross product calculations. To apply the right-hand rule, point your thumb in the direction of the first vector (velocity), and curl your fingers toward the second vector (magnetic field). The direction your palm faces or your thumb points, after executing this maneuver, shows the direction of the resultant vector.

• Thumb: direction of velocity \( \vec{v} \)
• Fingers: curl in the direction of the magnetic field \( \vec{B} \)
• Palm/thumb: direction of induced emf or force

In this exercise, this rule helps determine the orientation of the induced electromotive force (emf) in the rod. By using the right-hand rule, we can visualize and confirm that the induced electric field points towards the \(-y\)-axis, indicating which end of the rod is at a higher potential.