Coulomb's Law to find the net force

[kiwikahuna]

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.

 

[Astronuc]

d is the only unknown.

One has F, from which one finds F_x = F cos (theta). What is cos (theta) in terms of 'd'?

How would one find the maximum of F_x as a function of d?

 

[kiwikahuna]

theta = adjacent/hypotenuse


How do you already know what F is?

 

[Astronuc]

My apology - I should have asked - What is cos (theta) in terms of 'd'?

Coulomb's law: F = kq1q2/r^2

which one then writes

F = 8.98755e9 Nm^2/C^2 * 8e-18 C * 9e-18 C / (4 + d^2)

 

[kiwikahuna]

cos theta would equal d/ sqrt(4 + d^2)?

Could you clarify a little bit more about how to solve this problem? Sorry I'm a bit confused.

 

[PFStudent]

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:

$$\vec{F}_{12} = \frac{k_{e}q_{1}q_{2}}{{r_{12}}^2}\hat{r}_{21}$$

Scalar Form:

$$|\vec{F}_{12}| = \frac{k_{e}|q_{1}||q_{2}|}{{r_{12}}^2}$$

Now, you also know that,

$$F_{21}_{x} = |\vec{F}_{21}|cos{\theta}$$

And you need to find the value of d that would maximize $F_{21}_{x}$, therefore consider rewriting as,

$$F_{21}_{x}(d) = |\vec{F}_{21}|\left(\frac{d}{\sqrt{d^2+2^2}}\right)$$

$$F_{21}_{x}(d) = \left(\frac{k_{e}|q_{1}||q_{2}|}{{r_{12}}^2}\right)\left(\frac{d}{\sqrt{d^2+2^2}}\right)$$

$$F_{21}_{x}(d) = \left(\frac{k_{e}|q_{1}||q_{2}|}{{(\sqrt{d^2+2^2})}^2}\right)\left(\frac{d}{\sqrt{d^2+2^2}}\right)$$

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