The electric potential at a point (x;y)
on the line segment extending
from (0;3) to (2;0) is given by
At what point on this segment is the potential a minimum?
(Hint: the equation of the line segment is )
Hi joey,
try using the hint to replace with
Then you will have a quadratic in x.
If you differentiate the quadratic with respect to x,
you can equate that to zero to discover which x causes the tangent
to the curve to be parallel to the x-axis.
Solve for the derivative=0
When you have found that x, use it to find the vertical co-ordinate corresponding to it on the line segment..
okay this is what i have so far
equate p'=0 therfore
hope i am right so far
thereafter i went on to think that the co ordinates given that the values of x=0 and x=2
when x=2
p'=
since the answer is >0 it is the minimum
to find the min value at x=2
the min values are (2:24),please tell me that is right,or i need serious help
You are examining the quadratic equation to find the value of x at the minimum of the quadratic equation.
The quadratic equation generates the curve for P.
You found the minimum x co-ordinate on that.
The original question is asking you to find the point on the line segment given
at which the voltage is a minimum.
Place the x you discovered, corresponding to minimum potential back into the line equation to discover the corresponding value of y.
Hi Joey,
I could say your answer is correct,
but I'd much rather ask
1. "Do you understand why we substitute y into the equation for P ?"
2. "Do you understand why we differentiate the resulting quadratic ?"
3. "Do you understand why we substitute the minimum x back into the line equation ?"
The reason I'd rather ask those questions is because when you understand
why you are taking those steps in that order,
you are then mastering the technique and you will confidently answer
such questions in the future.
You will get the right answer (barring blunders) if you understand the technique.
Yes, the answer you got is correct.
If you try working it through again from start to finish, it will be worthwhile.
You could also find the minimum x without calculus, since quadratic curves are symmetrical about the turning point.
To find the other value of x making P=18
The axis of symmetry which goes vertically through the minimum
lies halfway between 0 and 2.4, which is x=1.2