E and B Fields

Electricity and Magnetism Level 3

At time t = 0 t = 0 , a particle of mass m = 1 m = 1 and charge q = + 1 q = +1 is at rest at the origin in the x y z xyz coordinate system. There are uniform electric and magnetic fields ( E ⃗ (\vec{E} and B ⃗ ) \vec{B}) throughout all of space.

E ⃗ = ( E x , E y , E z ) = ( 0 , 1 , 0 ) B ⃗ = ( B x , B y , B z ) = ( 0 , 0 , 1 ) \vec{E} = (E_x, E_y, E_z) = (0,1,0) \\ \vec{B} = (B_x, B_y, B_z) = (0,0,1)

If the spatial coordinates of the particle at time t = 1 t = 1 are ( x f , y f , z f ) (x_f, y_f, z_f) , enter your answer as x f + y f + z f x_f + y_f + z_f


The answer is 0.6182.

Steven Chase by Steven Chase , 37, USA

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1 solution

The equations of motion of the particle are d v x d t = v y \dfrac{dv_x}{dt}=v_y , d v y d t = 1 − v x \dfrac{dv_y}{dt}=1-v_x and d v z d t = 0 \dfrac{dv_z}{dt}=0 . Solving these equations and using initial conditions we get x f = t − s i n t x_f=t-sint , y f = 1 − c o s t y_f=1-cost and z = 0 z=0 at t = 1 t=1 . So x f + y f + z f = 2 − s i n 1 − c o s 1 = 0.6182... x_f+y_f+z_f=2-sin1-cos1=0.6182...

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