Question about relationship between Acceleration and Electric Potential

Sep 2019
1
0
Hi everyone. Im a Engineering student and i having doubts about this question:

"What is the acceleration vector \(\displaystyle a(x,y,z)\) of a particle of mass \(\displaystyle m0\) and charge \(\displaystyle q0\) when there is a Potential vector\(\displaystyle V(x,y,z) = c0 x3 c1 + x-5 y5 z4\) where '\(\displaystyle c0\)' and '\(\displaystyle c1\)' are constants."

Well I started with the relation

\(\displaystyle V = -W/q0 \) (1)

and \(\displaystyle W = F • d\) (being F and d vectors) (2)

also \(\displaystyle F = m0*a\) (being 'a' a vector) (3)

So I ended up with this equation that relates a and V replacing equation (3) in (2):

\(\displaystyle W = (m0*a) • d \)(4)

And replacing (4) in (1) I have:

\(\displaystyle V = - m0*a*d / q0 \)(5)

Which means:

\(\displaystyle a = - V*q0 / m0*d\) (6)

Here I got stuck and don't know how to proceed. I appreciate very much any help.
 
Oct 2017
523
236
Glasgow
There's no need to introduce work done here. The potential is related to the electric field strength by:

\(\displaystyle \vec{E} = - \nabla V\)

Assuming the scalar field \(\displaystyle V(x,y,z) = c_0 c_1 x^3 + x - 25 yz^4\), we can compute the gradient, \(\displaystyle \nabla V\).

\(\displaystyle \nabla V = \frac{\partial V}{\partial x} \hat{x} + \frac{\partial V}{\partial y} \hat{y} + \frac{\partial V}{\partial z} \hat{z}\)
\(\displaystyle = \left(3 c_0 c_1 x^2 +1\right) \hat{x} - 25z^4 \hat{y} - 100 y z^3 \hat{z}\)

Assuming that the only force in the problem is the electric force, we have

\(\displaystyle \vec{F} = \vec{E} q_0 = m_0 \vec{a}\).

Therefore,

\(\displaystyle \vec{a} = \frac{\vec{E} q_0}{m_0} = \frac{-q_0 \nabla V }{m_0}\)
\(\displaystyle = \frac{q_0}{m_0} \left(-\left(3 c_0 c_1 x^2 +1\right) \hat{x} + 25z^4 \hat{y} + 100 y z^3 \hat{z}\right)\)

You might want to double-check that the potential you wrote down is the same as the one I assumed...
 
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