Force from Potentials

Electricity and Magnetism Level 3

The electric field E ⃗ \vec{E} and magnetic flux density B ⃗ \vec{B} can generally be written in terms of a scalar electric potential V V and a vector magnetic potential A ⃗ \vec{A} .

E ⃗ = − ∇ V − ∂ A ⃗ ∂ t B ⃗ = ∇ × A ⃗ \vec{E} = -\nabla V - \frac{\partial{\vec{A}}}{\partial{t}} \\ \vec{B} = \nabla \times \vec{A}

Suppose the potentials are the following (depending on three space variables and one time variable):

V ( x , y , z , t ) = x t + y 2 + z 2 t 2 A ⃗ ( x , y , z , t ) = ( A x , A y , A z ) = ( x 2 y , z + t , y z t ) V(x,y,z,t) = x t + y^2 + z^2 t^2 \\ \vec{A}(x,y,z,t) = (A_x, A_y, A_z) = (x^2 y, z + t, y z t)

There is a particle with charge q = + 1 q = +1 . At time t = 2 t = 2 , the particle's position and velocity are:

( x , y , z ) = ( 4 , 1 , 2 ) ( v x , v y , v z ) = ( − 1 , 3 , 2 ) (x,y,z) = (4,1,2) \\ (v_x, v_y, v_z) = (-1,3,2)

What is the magnitude of the force on the particle at that time?

Bonus: The beginning of the "classical gauge theory" section of this wikipedia page is an interesting read.


The answer is 58.29.

Steven Chase by Steven Chase , 37, USA

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1 solution

B ⃗ \vec B not magnetic field , but magnetic flux density or magnetic induction .

E ⃗ = − ∇ ⃗ V − ∂ A ⃗ ∂ t = − ( i ^ t + j ^ ( 2 y + 1 ) + k ^ ( 2 z t 2 + y z ) ) \vec E=-\vec \nabla V-\dfrac{\partial \vec A}{\partial t}=-\left (\hat i t+\hat j (2y+1)+\hat k (2zt^2+yz)\right ) .

Substituting values, E ⃗ = − ( 2 i ^ + 3 j ^ + 18 k ^ ) \vec E=-(2\hat i+3\hat j+18\hat k) .

B ⃗ = ∇ ⃗ × A ⃗ = i ^ ( z t − 1 ) − k ^ x 2 \vec B=\vec \nabla \times \vec A=\hat i (zt-1)-\hat k x^2

Substituting values B ⃗ = 3 i ^ − 16 k ^ ⟹ v ⃗ × B ⃗ = − 48 i ^ − 10 j ^ − 9 k ^ \vec B=3\hat i-16\hat k\implies \vec v\times \vec B=-48\hat i-10\hat j-9\hat k

So, force F ⃗ = q ( E ⃗ + v ⃗ × B ⃗ ) = − ( 50 i ^ + 13 j ^ + 27 k ^ ) ⟹ ∣ F ⃗ ∣ = 5 0 2 + 1 3 2 + 2 7 2 ≈ 58.292366 \vec F=q(\vec E+\vec v\times \vec B) =-(50\hat i+13\hat j+27\hat k) \implies |\vec F|=\sqrt {50^2+13^2+27^2}\approx \boxed {58.292366} .

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