Coulomb's Law to find the net force

  • Thread starter kiwikahuna
  • Start date
  • #1
61
0

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.
 

Answers and Replies

  • #2
Astronuc
Staff Emeritus
Science Advisor
18,932
2,258
d is the only unknown.

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

How would one find the maximum of Fx as a function of d?
 
Last edited:
  • #3
61
0
theta = adjacent/hypotenuse


How do you already know what F is?
 
  • #4
Astronuc
Staff Emeritus
Science Advisor
18,932
2,258
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)
 
  • #5
61
0
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.
 
  • #6
170
0

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,

[itex]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[/itex]

[itex]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[/itex]

Also let the distance between [itex]q_{1}[/itex] and [itex]q_{2}[/itex] be [itex]r_{12}[/itex] (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:

[tex]
\vec{F}_{12} = \frac{k_{e}q_{1}q_{2}}{{r_{12}}^2}\hat{r}_{21}
[/tex]

Scalar Form:

[tex]
|\vec{F}_{12}| = \frac{k_{e}|q_{1}||q_{2}|}{{r_{12}}^2}
[/tex]

Now, you also know that,

[tex]
F_{21}_{x} = |\vec{F}_{21}|cos{\theta}
[/tex]

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

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

[tex]
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)
[/tex]

[tex]
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)
[/tex]

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.

Best,

-PFStudent
 
Last edited:

Related Threads on Coulomb's Law to find the net force

Replies
1
Views
7K
  • Last Post
Replies
0
Views
4K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
3
Views
3K
Replies
2
Views
6K
  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
5K
Replies
3
Views
18K
Top