A particle with a charge of -2.50 ×10^-8 C is moving with an instantaneous velocity of magnitude

step 1 Okay, so we know that the magnetic force is going to be equal. Magnetic force on a charged parti

A particle with a charge of -2.50 ×10^-8 C is moving with an...

A particle with a charge of $-2.50 \times 10^{-8} \mathrm{C}$ is moving with an instantaneous velocity of magnitude 40.0 $\mathrm{km} / \mathrm{s}$ in the $x y-$ plane at an angle of $50^{\circ}$ counterclockwise from the $+x$ axis. What are the magnitude and direction of the force exerted on this particle by a magnetic field with magnitude 2.00 T in the (a) $-x$ direction, and (b) $+z$ direction?

A particle with a charge of $-2.50 \times 10^{-8} \mathrm{C}$ is moving with an instantaneous velocity of magnitude 40.0 $\mathrm{km} / \mathrm{s}$ in the $x y-$ plane at an angle of $50^{\circ}$ counterclockwise from the $+x$ axis. What are the magnitude and direction of the force exerted on
this particle by a magnetic field with magnitude 2.00 T in the (a) $-x$ direction, and (b) $+z$ direction?

Key Concepts

Charge Sign Effect

The sign of the charge plays a crucial role in determining the direction of the magnetic force. While the mathematical procedure using the cross product yields a direction, a negative charge reverses this direction. This concept is vital in predicting motion, especially in systems like particle accelerators or in the Earth's magnetosphere.

Right-Hand Rule

The right-hand rule is a mnemonic for determining the direction of the cross product vector in three-dimensional space. When applying it to the magnetic force, one points the fingers in the direction of the velocity vector and curls them toward the magnetic field vector; the thumb then points in the direction of the force. However, if the charge is negative, the force is in the opposite direction to that indicated by the right-hand rule.

Vector Decomposition in Cartesian Coordinates

Decomposing vectors into their Cartesian components involves breaking down a vector into mutually perpendicular components, typically along the x, y, and z axes. This is essential when dealing with forces and velocities in multiple dimensions, allowing the calculation of individual force components and the overall resultant force in problems involving magnetic fields and moving charges.

Lorentz Force

The Lorentz force is the net force experienced by a charged particle moving in electric and magnetic fields. For a purely magnetic force, it is given by the equation F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. This concept is foundational in electromagnetism as it explains how charged particles are deflected or accelerated by magnetic fields.

Vector Cross Product

The cross product is a mathematical operation on two vectors that results in a third vector perpendicular to the plane formed by the original vectors. In the context of the magnetic force on a moving charge, the magnitude of the force is given by F = |q| v B sin?, where ? is the angle between the velocity and magnetic field vectors, emphasizing that the force depends on both the magnitude and the relative orientation of v and B.

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