Maximizing Electrostatic Force between Two Point Charges

[Vishakha]

Homework Statement

Two point charges q and λq located at the points, x=a & x=μa respectively. If the sum of the two charges is constant,what is the value of λ for which the magnitude of the electrostatic force is maximum?

Homework Equations

The Attempt at a Solution

For force to be maximum dF/dq =0 and d^F/dq^2 <0. When I tried to calculate dF/dq =0 I got λ=0.

 

[BvU]

Well, how did you do that? Can you post your steps in detail ?

By the way: why calculate $dF\over dq$ ?

 

[Vishakha]

Well, how did you do that? Can you post your steps in detail ?

By the way: why calculate $dF\over dq$ ?

I already wrote that in The attempt at a solution section.

F= λq²/4πε₀ (μa-a)²
⇒ dF/dq = 2λq/4πε₀(μa-a)² =0
⇒ 2λq = 0
q≠0 ⇒λ=0

 

[BvU]

If you want to find the maximum of a function of x you take the derivative wrt x.
Here, do you want to consider the force as a function of q ?

Note also that you forgot to make use of the given that

the sum of the two charges is constant

 

[Vishakha]

If you want to find the maximum of a function of x you take the derivative wrt x.
Here, do you want to consider the force as a function of q ?

Note also that you forgot to make use of the given that

If I used q+λq = C ⇒ λq=C-q
Then 2λq=0 ⇒ 2(C-q) = 0 ⇒q=C

Let F is function of distance

F = λq²/4πε₀a²(μ-1)²
⇒ -2λq²/4πε₀a³(μ-1)² = 0
⇒ -2λq² = 0
⇒ λq= 0 or λq=C

 

[BvU]

Are you saying $\lambda = 0 \Rightarrow F = 0$ too ? That is easily proven wrong !

 

[Vishakha]

Are you saying $\lambda = 0 \Rightarrow F = 0$ too ? That is easily proven wrong !

You are right. F shouldn't be zero but I don't find any mistake in my calculation.

 

[BvU]

I don't find any mistake in my calculation

I did and I tried to point it out. You want to express F in terms of $\lambda$ and differentiate wrt $\lambda$. Make a start ...

 

[Vishakha]

I did and I tried to point it out. You want to express F in terms of $\lambda$ and differentiate wrt $\lambda$. Make a start ...

You mean I have to differentiate F wrt λ and distance between charges and q is constant.

 

[BvU]

Yes and No. In that order:

the sum of the two charges is constant

 

[Vishakha]

Yes and No. In that order:

λ+qλ = C
⇒dq = -(q+1)dλ/λ _... (1)

dF/da = [(μa-a)² { 2λq dq + q² dλ} - 2q²λa (μ-1)² ]/ 4πε₀ (μa-a)² = 0

After putting value of eq (1) I got final eq
-q(q+1) dλ = 2λa

 

[BvU]

Can you explain why you are now differentiating wrt a ? I thought we agreed to seek a value for $\lambda$ that gives the maximum $F$ ?

And: do you think we can leave out the constants $\displaystyle {1\over 4\pi\varepsilon_0 (\mu a - a)^2 }$ ?

 

[Vishakha]

And: do you think we can leave out the constants $\displaystyle {1\over 4\pi\varepsilon_0 (\mu a - a)^2 }$ ?

I got λ=1 if we leave the constants.

But I don't understand why are we differentiating wrt λ?

 

[gneill]

I got λ=1 if we leave the constants.

But I don't understand why are we differentiating wrt λ?

Because the problem asks, "what is the value of λ for which the magnitude of the electrostatic force is maximum?"

 

[BvU]

I got λ=1 if we leave the constants.

But I don't understand why are we differentiating wrt λ?

Can you explain how you found $\lambda = 1$ ?
And can you justify leaving out the constants ? Why is that allowed ?