A particle of mass m and charge +q is shot with velocity 𝐯 into a region of uniform magnetic fie

step 1 we have the centipatal force mv square over r is given by q e minus q v b so we have this is b n

A particle of mass m and charge +q is shot with velocity 𝐯...

A particle of mass $m$ and charge $+q$ is shot with velocity $\mathbf{v}$ into a region of uniform magnetic field $\mathbf{B}$. Suppose that $\mathbf{v}$ points to the top of the page and $\mathbf{B}$ points out of the page. Ignoring gravity, the particle (A) travels in a straight line. (B) moves clockwise in a circle with radius $r=q B / m v$. (C) moves counterclockwise in a circle with radius $r=q B / m v$. (D) moves clockwise in a circle with radius $r=m v / q B$. (E) moves counterclockwise in a circle with radius $r=m v / q B$.

A particle of mass $m$ and charge $+q$ is shot with velocity $\mathbf{v}$ into a region of uniform magnetic field $\mathbf{B}$. Suppose that $\mathbf{v}$ points to the top of the page and $\mathbf{B}$ points out of the page. Ignoring gravity, the particle
(A) travels in a straight line.
(B) moves clockwise in a circle with radius $r=q B / m v$.
(C) moves counterclockwise in a circle with radius $r=q B / m v$.
(D) moves clockwise in a circle with radius $r=m v / q B$.
(E) moves counterclockwise in a circle with radius $r=m v / q B$.

Key Concepts

Lorentz Force

The Lorentz force describes the force experienced by a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), where 'q' is the charge of the particle, 'v' is its velocity vector, and 'B' is the magnetic field vector. This concept is fundamental in electromagnetism as it explains how charged particles are deflected by magnetic fields.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the force acting on a charged particle in a magnetic field. By pointing the fingers in the direction of the velocity and curling them toward the magnetic field, the thumb indicates the direction of the force (for a positive charge). This intuitive tool helps predict motion in magnetic fields without performing vector cross-product calculations.

Circular Motion due to Magnetic Fields

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts continuously perpendicular to its velocity, providing the centripetal force necessary for circular motion. The constant magnitude and perpendicular direction of the force result in uniform circular motion, with the radius determined by the balance of the magnetic force and the centripetal force requirement.

Centripetal Force and Radius of Circular Motion

In the context of magnetic fields, the magnetic force serves as the centripetal force that keeps the particle moving in a circular path. The radius of this path can be derived from the equilibrium between the magnetic force (qvB) and the centripetal force requirement (mv²/r), leading to the radius being expressed as r = m*v/(q*B). This relationship is key to analyzing the motion of charged particles in magnetic fields.

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