Magnetic Fields & Cross Products

[Soaring Crane]

I desperately need help, for I am quite lost on this question:

Consider the example of a positive charge moving in the xy plane with velocity v = cos(theta)x + sin(theta)y (i.e., at angle theta with respect to the x axis). If the local magnetic field is in the +z direction, what is the direction of the magnetic force acting on the particle?

Express the direction of the force in terms of theta, as a linear combination of unit vectors, x, y, and z.

F = qBV

One hint given was to find the determinant expression for the cross product, but I don't know what this means.

Thanks for any replies.

 

[dextercioby]

$$\vec{F}_{Lorentz}=q \vec{v}\times\vec{B}$$

Do you know to compute a cross product...?Through what methods...?


Daniel.

 

[thepatientmental]

The cross product, also known as the vector product, is a mathematical operation that takes two vectors and produces a new vector that is perpendicular to both of the original vectors. In this case, we are dealing with a charge moving in a magnetic field, which means we need to use the cross product to find the direction of the magnetic force acting on the particle.

To find the direction of the magnetic force, we can use the formula F = qv x B, where q is the charge of the particle, v is its velocity, and B is the magnetic field. In this case, we can rewrite the formula as F = q(v x B), where v and B are both vectors.

To find the cross product, we need to use a determinant expression. The determinant expression for the cross product is:

v x B = |i j k|
|vx vy vz|
|Bx By Bz|

where i, j, and k are unit vectors in the x, y, and z directions respectively, and vx, vy, vz are the components of the velocity vector v, and Bx, By, Bz are the components of the magnetic field vector B.

In our case, we have v = cos(theta)x + sin(theta)y and B = 0x + 0y + Bz. Plugging these values into the determinant expression, we get:

v x B = |i j k|
|cos(theta) sin(theta) 0|
|0 0 Bz|

Now, to find the direction of the magnetic force, we need to evaluate this determinant. The result will be a vector in the direction of the force. To do this, we can use the "right-hand rule", which states that if you curl your fingers from the first vector (v) to the second vector (B), your thumb will point in the direction of the cross product (F).

In this case, if we curl our fingers from v to B, our thumb will point in the positive z direction. This means that the direction of the magnetic force is in the positive z direction.

To express this direction in terms of unit vectors, we can write F = Bz(cos(theta)i + sin(theta)j). This means that the magnetic force is a combination of the x and y unit vectors, with coefficients determined by the angle theta and the strength of the magnetic field Bz.

I hope this