Maximizing Potential: Solving for a in a Function with Two Particle Charges

[the-rocketman]

I have this function (to calculate potential for classical over-barrier model if you're wondering) where qp and qt are both (different) particle charges:

V(a,R) = -(qp /R) - [qt / (R-a)]

What is dV/da?

The maximum of the function occurs when dV/da = 0, and I need to solve for a. So does solving for a yield R[1 + sqrt(qt/qp)]^-1?

Thanks

 

[Brad Barker]

I have this function (to calculate potential for classical over-barrier model if you're wondering) where qp and qt are both (different) particle charges:

V(a,R) = -(qp /R) - [qt / (R-a)]

What is dV/da?

use the chain rule.

 

[the-rocketman]

Hm... but it has 2 variables. I haven't had any multivariable calc.

 

[sic]

what?
try to sketch the function. obviously, the function doesn't have a maximum for any constant R.
what is the question from the beginning?

 

[the-rocketman]

R and a are not constants, they are the variables. the constants are the q values

 

[hypermorphism]

The maximum of the function occurs when dV/da = 0...

That isn't necessarily true for multivariable functions. Instead, you need to see when the gradient vector [dV/da, dV/dR] = 0 and use the discriminant to tell what type of critical point it is.

 

[Brad Barker]

Hm... but it has 2 variables. I haven't had any multivariable calc.

the process is the same. just imagine that R isn't a variable, so long as you want to find dV/da.

(the "d's" should be curly, like so: $$\partial$$)