The magnetic force, always acting perpendicular to a charged particle's velocity, does no work on the particle, thus it cannot change the particle's speed or kinetic energy. This perpendicularity, determined by the cross product, only alters the particle's direction, causing it to follow curved or circular paths. Conversely, an electric field can exert a force parallel to the particle's motion, thereby performing work and changing the particle's speed and kinetic energy. For instance, particle accelerators utilize electric fields for increasing particle speed and magnetic fields for controlling the direction of the particle beam.
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$\gg$Mathematical Structures Underlying Physical Laws
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The most crucial concept is that the magnetic force ( $F_B$ ) is always perpendicular to the particle's velocity ( $v$ ). This is a defining characteristic of the magnetic force, which is determined by the cross-product ( $v \times B$ ).
Since work is defined as the component of force in the direction of motion, and the magnetic force has zero component in the direction of motion, no work can be done. Mathematically, this is shown by the dot product: $d W=F_B \cdot d x=0$. Because the force vector is perpendicular to the displacement vector at every instant, their dot product is always zero.
Because no work is done, a magnetic field cannot change the kinetic energy or speed of a charged particle. It can only change the particle's direction, causing it to move in a curved or circular path. It is the electric field that can accelerate the particle and thus change its speed and kinetic energy.
while both electric and magnetic fields can exert a force on a charged particle, only the electric field does work. The magnetic force is always perpendicular to the particle's direction of motion, meaning it can change the particle's path but cannot change its kinetic energy. This is why the "Total Work Done" only increases when the electric field is turned on.
3D Lorentz Force and Work Done
3D Lorentz Force and Work Done
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Proving the Cross Product Rules with the Levi-Civita Symbol
Proving the Epsilon-Delta Relation and the Bac-Cab Rule
Simplifying Levi-Civita and Kronecker Delta Identities
Why a Cube's Diagonal Angle Never Changes
How the Cross Product Relates to the Sine of an Angle
Finding the Shortest Distance and Proving Orthogonality for Skew Lines
A Study of Helical Trajectories and Vector Dynamics
The Power of Cross Products: A Visual Guide to Precessing Vectors
Divergence and Curl Analysis of Vector Fields
Unpacking Vector Identities: How to Apply Divergence and Curl Rules
Commutativity and Anti-symmetry in Vector Calculus Identities
Double Curl Identity Proof using the epsilon-delta Relation
The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
Surface Parametrisation and the Verification of the Gradient-Normal Relationship
Proof and Implications of a Vector Operator Identity
Conditions for a Scalar Field Identity
Solution and Proof for a Vector Identity and Divergence Problem
Kinematics and Vector Calculus of a Rotating Rigid Body
Work Done by a Non-Conservative Force and Conservative Force
The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
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