A particle with charge -5.60 nC is moving in a uniform magnetic field B⃗=-(1.25 T) k̂ . The magn

A particle with charge -5.60 nC is moving in a uniform...

A particle with charge $-5.60 \mathrm{nC}$ is moving in a uniform magnetic field $\vec{\boldsymbol{B}}=-(1.25 \mathrm{T}) \hat{\boldsymbol{k}}$ . The magnetic force on the particle is measured to be $\vec{\boldsymbol{F}}=-\left(3.40 \times 10^{-7} \mathrm{N}\right) \hat{\imath}+(7.40 \times$ $10^{-7} \mathrm{N} ) \hat{J}$ . (a) Calculate all the components of the velocity of the particle that you can from this information. (b) Are there components of the velocity that are not determined by the measurement of the force? Explain. (c) Calculate the scalar product $\vec{v} \cdot$ What is the angle between $\vec{v}$ and $\vec{F} ?$

Key Concepts

Underdetermined Velocity Components

When a charged particle moves in a magnetic field, the component of its velocity that is parallel to the magnetic field does not affect the magnetic force, because the cross product of parallel vectors is zero. Consequently, measurements of the magnetic force only determine the perpendicular components of the velocity, leaving the parallel component undetermined.

Orthogonality in Magnetic Force

Due to the nature of the cross product, the magnetic force is always perpendicular to both the velocity and the magnetic field. This orthogonality implies that any component of the velocity that is parallel to the magnetic field does not contribute to the force, leading to an indeterminate situation where such components cannot be deduced from the force measurement alone.

Scalar (Dot) Product and Angle Calculation

The scalar or dot product of two vectors provides a means to calculate the angle between them, given by the formula v · F = |v||F|cos(?). In problems involving magnetic forces, this concept is used to determine the geometric relationship between the velocity and the force vectors, even though the force is perpendicular to part of the velocity vector.

Lorentz Force Law

The Lorentz force law is fundamental in electromagnetism and describes the force acting on a charged particle moving in a magnetic field. It is given by F = q(v × B), which means that the force is the cross product of the velocity and the magnetic field scaled by the charge. This law is critical for analyzing how charged particles behave in magnetic environments.

Vector Cross Product

The vector cross product is an operation on two vectors that results in a third vector perpendicular to the plane containing the original vectors. In the context of the Lorentz force, this property indicates that the magnetic force is orthogonal to both the velocity of the charged particle and the magnetic field, which helps in understanding what components of the velocity contribute to the force.

Transcript

00:01 We have a problem here that's going through a genetic force on a charged particle.

00:08 In this problem, we have some charged particle, which has a charge of negative 5 .6 nanopulums, going through a b field expressed here, has 1 .25 teslas in the negative c direction, and it has a force, which has this force vector expressed here.

00:29 And what we want to know for the first part is what is the velocity of the particle that gives us this force.

00:38 So to figure this out, we will use the equation for force on a moving charged particle in a magnetic field.

00:46 That is f is equal to qv cross b.

00:52 So the cross product between the velocity of the particle and the magnetic field.

00:57 So we'll just kind of express this differently.

01:03 So q, you can write this like this, magnitude of the field b.

01:11 And then we'll express v as its components.

01:15 So we have the x component of velocity, plus y component, plus the z component of velocity.

01:26 And this will be crossed with the direction of b, which we know is in the z direction.

01:32 Or k hat.

01:36 Now let's just distribute this k through it, right? so this is the same.

01:42 And now we have vx times the direction of i cross k plus v y, j cross k, and then this last part is k cross k.

02:01 Okay.

02:03 Now we want to go through these cross products.

02:05 So the first one, we should know right away is that this last one is equal to zero, right? because the cross product of a vector and itself is equal to zero.

02:18 Now the other one, we'll cry a little bit more effort.

02:22 So if i cross k here, we can figure out by just using a cross product, or a right hand rule, right? so we have axes that look like this.

02:34 So the up direction here is z, this would be x, and that is y.

02:39 So these are our directions.

02:41 So to get this cross product out, we would point our fingers in x direction, wrap our fingers around in the z direction, and then the way that our thumb points points towards the direction of this cross product, which would be in the negative y direction...

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