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A charged particle q moving with a velocity \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v} \) in a magnetic field \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over B} \) experiences a force given by \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over F} = q\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v} \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over B} } \right).\) What is the work done by the force during a displacement \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over{\Delta r}}\) of the charged particle?

A charged particle q moving with a velocity \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v} \) in a magnetic field \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over B} \) experiences a force given by \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over F} = q\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v} \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over B} } \right).\) What is the work done by the force during a displacement \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over{\Delta r}}\) of the charged particle? Correct Answer Zero

Concept:

  • Lorentz force: The Lorentz force equation describes the magnitude of the force that a moving electric charge would feel as a result of being in the presence of a magnetic field B̅ and an Electric field E̅.

i.e.

  • According to Lorentz’s force when the charge is moving in presence of both an electric and magnetic field, the force acting on it will be the sum of the two forces, i.e.
  • Fnet = F̅elec. + F̅mag. 
  • Fnet = q × E̅ + q (v̅ × B̅)
  • Fnet = q (E̅ + v̅ × B̅)

 

q = charge of the particle

E̅ = Electric field vector

v̅ = velocity vector of the particle

B̅ = magnetic field vector

Work 

  • Work is said to be done by a force when the body is displaced actually through some distance in the direction of the applied force.
  • Since body is being displaced in the direction of F, therefore work done by the force in displacing the body through a distance s is given by

W = F→⋅s→ W = F→ ⋅s→

Or, W = Fs cosθ

  • Thus work done by a force is equal to the scalar or dot product of the force and the displacement of the body.

Calculation:

Lorentz force = magnetic force + electric force 

F =

F = q (V × B) {F is perpendicular to both V and B. (Due to cross product)}

If ds→ is the instantaneous displacement of the change 

ds→ is also perpendicular to F

W = F.ds

W = Fs cos 90° {cos 90° = 0}

W = 0 

The work done by the force is zero.

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