kiwikahuna said:
Homework Statement
Charge 8e-18 C is on the y-axis a distance 2 m from the origin and charge
9e-18 C is on the x-axis a distance d from the origin. The Coulomb constant is 8.98755e9 Nm^2/C^2.
What is the value of d for which the x component of the force on 9e-18 C is the greatest?
Homework Equations
Coulomb's law: F = kq1q2/r^2
The Attempt at a Solution
I tried to use Coulomb's law to find the net force and then to find the force in the x direction but I became very stuck.
F = 8.98755e9 Nm^2/C^2 * 8e-18 C * 9e-18 C / (4 + d^2)
My problem is I have two unknowns and I can't find the value of d. Please help if you can.
Hey,
Let,
q_{1} = 8{\textcolor[rgb]{1.00,1.00,1.00}{.}}x{\textcolor[rgb]{1.00,1.00,1.00}{.}}10^{-18}{\textcolor[rgb]{1.00,1.00,1.00}{.}}C
q_{2} = 9{\textcolor[rgb]{1.00,1.00,1.00}{.}}x{\textcolor[rgb]{1.00,1.00,1.00}{.}}10^{-18}{\textcolor[rgb]{1.00,1.00,1.00}{.}}C
Also let the distance between q_{1} and q_{2} be r_{12} (read as: distance r from 1 to 2) instead of plain r, makes the problem clearer.
First, draw a picture, makes the problem much easier.
Second, consider what you already know.
You know Coulomb's Law:
Vector Form:
<br />
\vec{F}_{12} = \frac{k_{e}q_{1}q_{2}}{{r_{12}}^2}\hat{r}_{21}<br />
Scalar Form:
<br />
|\vec{F}_{12}| = \frac{k_{e}|q_{1}||q_{2}|}{{r_{12}}^2}<br />
Now, you also know that,
<br />
F_{21}_{x} = |\vec{F}_{21}|cos{\theta}<br />
And you need to find the value of d that would maximize <br />
F_{21}_{x}, therefore consider rewriting as,
<br />
F_{21}_{x}(d) = |\vec{F}_{21}|\left(\frac{d}{\sqrt{d^2+2^2}}\right)<br />
<br />
F_{21}_{x}(d) = \left(\frac{k_{e}|q_{1}||q_{2}|}{{r_{12}}^2}\right)\left(\frac{d}{\sqrt{d^2+2^2}}\right)<br />
<br />
F_{21}_{x}(d) = \left(\frac{k_{e}|q_{1}||q_{2}|}{{(\sqrt{d^2+2^2})}^2}\right)\left(\frac{d}{\sqrt{d^2+2^2}}\right)<br />
Now ask yourself, "Given a function of a single variable, how do you maximize that function? (hint: think calculus)".
Also remember d is a variable, not a constant.
-PFStudent
