Force on Moving Charges in a Magnetic Field

In summary, the problem involves a positive charge moving in the xy plane with a given velocity and a magnetic field in the +z direction. To find the direction of the magnetic force on the particle, the cross product equation for the Lorentz force can be applied. The direction of the force can be expressed as a linear combination of unit vectors, specifically in terms of \theta and the vectors \hat{x}, \hat{y}, and \hat{z}.
  • #1
Number1Ballar
14
0

Homework Statement


Consider the example of a positive charge q moving in the xy plane with velocity [tex]\hat{v}[/tex] = vcos([tex]\theta[/tex])[tex]\hat{x}[/tex] + vsin([tex]\theta[/tex])[tex]\hat{y}[/tex] (i.e., with magnitude v at angle [tex]\theta[/tex] 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 [tex]\theta[/tex], as a linear combination of unit vectors, [tex]\hat{x}[/tex], [tex]\hat{y}[/tex], [tex]\hat{z}[/tex].


Homework Equations


Cross product.
[tex]\vec{C}[/tex] = [tex]\vec{A}[/tex] X [tex]\vec{B}[/tex] = (AxBy - AyBx)[tex]\hat{z}[/tex] + (AyBz - AzBy)[tex]\hat{x}[/tex] + (AzBx - AxBz)[tex]\hat{y}[/tex]


The Attempt at a Solution


I don't know what to plug in, and overall am just confused as to what the formula means.

Any explanation would be greatly appreciated!
 
Physics news on Phys.org
  • #2
You only need to to apply the equation for the Lorentz force here. What have you learned about how to evaluate the cross product of two vectors?
 
  • #3


The formula you have provided is for the cross product, which is used to calculate the vector product of two vectors. In this case, we are interested in finding the direction of the magnetic force acting on a moving charge in a magnetic field. The formula for this force is given by F = q(\vec{v} X \vec{B}), where q is the charge of the particle, \vec{v} is its velocity, and \vec{B} is the magnetic field.

In this example, the charge is moving in the xy plane with a velocity given by \hat{v} = vcos(\theta)\hat{x} + vsin(\theta)\hat{y}. This means that the velocity vector is pointing in the direction of the angle \theta with respect to the x-axis. The magnetic field is in the +z direction, which means that the magnetic field vector is pointing upwards in the z direction.

To find the direction of the magnetic force, we can use the right-hand rule. If you point your right thumb in the direction of the velocity vector \vec{v} and your fingers in the direction of the magnetic field vector \vec{B}, your palm will be facing in the direction of the magnetic force vector \vec{F}.

In this case, since the velocity vector is in the xy plane and the magnetic field is in the +z direction, the magnetic force will be perpendicular to both of these vectors. This means that the magnetic force vector will be in the direction of the cross product \vec{v} X \vec{B}. Using the formula you provided, we can calculate the cross product as \vec{v} X \vec{B} = (vcos(\theta)\hat{x} + vsin(\theta)\hat{y}) X \hat{z} = (vsin(\theta))\hat{x} - (vcos(\theta))\hat{y}. This means that the direction of the magnetic force can be expressed as a linear combination of unit vectors \hat{x} and \hat{y}, with coefficients vsin(\theta) and -vcos(\theta), respectively.

In summary, the direction of the magnetic force acting on a positive charge moving in the xy plane with velocity \hat{v} = vcos(\theta)\hat{x} + vsin(\theta)\hat{y} in a magnetic field pointing in the +z direction can be expressed as F
 

What is the force on a moving charge in a magnetic field?

The force on a moving charge in a magnetic field is given by the equation F = qvB sinθ, where q is the charge, v is the velocity of the charge, B is the magnetic field, and θ is the angle between the velocity and the magnetic field.

How does the direction of the force on a moving charge in a magnetic field depend on the velocity and magnetic field?

The direction of the force on a moving charge in a magnetic field is always perpendicular to both the velocity and the magnetic field. The direction of the force can be determined using the right-hand rule, where the thumb points in the direction of the velocity, the fingers point in the direction of the magnetic field, and the palm points in the direction of the force.

What is the relationship between the force on a moving charge and the strength of the magnetic field?

The force on a moving charge is directly proportional to the strength of the magnetic field. This means that the stronger the magnetic field, the greater the force on the charge will be. This relationship is described by the equation F ∝ B.

How does the force on a moving charge change when the charge's velocity or the magnetic field is increased?

If the velocity of the charge is increased, the force will also increase proportionally. However, if the magnetic field is increased, the force will increase at a greater rate, as it is directly proportional to the strength of the magnetic field. This relationship is described by the equation F ∝ vB.

What is the significance of the angle between the velocity and the magnetic field in determining the force on a moving charge?

The angle between the velocity and the magnetic field, θ, is an important factor in determining the force on a moving charge. The force is greatest when the angle is 90 degrees (sinθ = 1), and is zero when the angle is 0 degrees (sinθ = 0). This means that the charge experiences the greatest force when it is moving perpendicular to the magnetic field, and no force when it is moving parallel to the magnetic field.

Similar threads

  • Introductory Physics Homework Help
Replies
25
Views
210
  • Introductory Physics Homework Help
Replies
3
Views
136
  • Introductory Physics Homework Help
Replies
1
Views
267
  • Introductory Physics Homework Help
Replies
1
Views
808
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
169
  • Introductory Physics Homework Help
Replies
3
Views
948
  • Introductory Physics Homework Help
Replies
25
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
818
Back
Top