A proton moves through a uniform magnetic field given by B⃗=(10 î-20 ĵ+30 k̂) mT. At time t1, th

step 1 In this question, we are given this situation. So we have a proton, okay, it has a velocity at T

A proton moves through a uniform magnetic field given by...

A proton moves through a uniform magnetic field given by $\vec{B}=(10 \hat{\mathrm{i}}-20 \hat{\mathrm{j}}+30 \hat{\mathrm{k}}) \mathrm{mT}$. At time $t_{1}$, the proton has a velocity given by $\vec{v}=v_{x} \hat{i}+v_{y} \hat{j}+(2.0 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{k}}$ and the magnetic force on the proton is $\vec{F}_{B}=\left(4.0 \times 10^{-17} \mathrm{~N}\right) \hat{\mathrm{i}}+\left(2.0 \times 10^{-17} \mathrm{~N}\right) \hat{\mathrm{j}} .$ At that instant, what are (a) $v_{x}$ and (b) $v_{y}$ ?

A proton moves through a uniform magnetic field given by $\vec{B}=(10 \hat{\mathrm{i}}-20 \hat{\mathrm{j}}+30 \hat{\mathrm{k}}) \mathrm{mT}$. At time $t_{1}$, the proton has a velocity
given by $\vec{v}=v_{x} \hat{i}+v_{y} \hat{j}+(2.0 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{k}}$ and the magnetic force on
the proton is $\vec{F}_{B}=\left(4.0 \times 10^{-17} \mathrm{~N}\right) \hat{\mathrm{i}}+\left(2.0 \times 10^{-17} \mathrm{~N}\right) \hat{\mathrm{j}} .$ At that
instant, what are (a) $v_{x}$ and (b) $v_{y}$ ?

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Key Concepts

Component Analysis in Vector Equations

Component analysis involves breaking vectors into their individual i, j, and k components to simplify and solve vector equations. This method is particularly useful in electromagnetic problems where the force, velocity, and magnetic field are expressed in terms of their directional components.

Uniform Magnetic Field

A uniform magnetic field is one in which the magnetic field vector has the same magnitude and direction at every point in space. This consistency simplifies analysis and calculations when determining the magnetic force on a moving charge.

Lorentz Force

The Lorentz force describes the force exerted on a charged particle moving through electric and magnetic fields. It is given by the equation F = q(v x B), indicating that the force is the product of the charge, the velocity, and the magnetic field, with the direction determined by the cross product of velocity and magnetic field.

Vector Cross Product

The vector cross product is a mathematical operation used to determine a vector that is perpendicular to two given vectors. Its magnitude is proportional to the sine of the angle between the two vectors, and its direction is determined by the right?hand rule, making it useful for determining forces in electromagnetic problems.

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