Find the unknown charges q1 and q2

[JessieS]

Homework Statement

The geometrical positions of point-like charges and point A situated in the xy-plane in terms of the length parameter a. The vector of electric field E at point A is shown schematically and measured as E = Exi + Eyj (that is, both Ex and Ey are given). If possible, find the unknown charges q1 and q2.

**E_2 is the electric field with respect to q2. Not shown in figure. **
**E_1 is the electric field with respect to q1. (Same direction as E) **

**k_e is Coulomb's constant.**

Homework Equations

E[/B] = Exi + Eyj
Ex = E_1*cos(45°) - E_2*cos(45°)
Ey = E_1*sin(45°) + E_2*sin(45°)
E = F/q = (k_e*q)/r^2

The Attempt at a Solution

E_1 [/B]= (k_e*q1)/(2a)^2 = (k_e*q1)/(4a^2)

E_2 = (k_e*q2)/(2a)^2 = (k_e*q2)/(4a^2)

Ex = (E_1 - E_2)*2/sqrt(2)

Ey = (E_1 + E_2)*2/sqrt(2)

E = 2/sqrt(2)*(E_1 - E_2) i + 2/sqrt(2)*(E_1 + E_2) j = 2/sqrt(2)*(k_e/(4a^2))*(q1 - q2) i + 2/sqrt(2)*(k_e/(4a^2))*(q1 + q2)I am stuck here; I'm not sure if I've been going about this correctly or what steps to take next.

Thank you.

 

[SammyS]

Homework Statement

The geometrical positions of point-like charges and point A situated in the xy-plane in terms of the length parameter a. The vector of electric field E at point A is shown schematically and measured as E = Exi + Eyj (that is, both Ex and Ey are given). If possible, find the unknown charges q1 and q2.

**E_2 is the electric field with respect to q2. Not shown in figure. **
**E_1 is the electric field with respect to q1. (Same direction as E) **

**k_e is Coulomb's constant.**

View attachment 102522

Homework Equations

E[/B] = Exi + Eyj
Ex = E_1*cos(45°) - E_2*cos(45°)
Ey = E_1*sin(45°) + E_2*sin(45°)
E = F/q = (k_e*q)/r^2

The Attempt at a Solution

E_1 [/B]= (k_e*q1)/(2a)^2 = (k_e*q1)/(4a^2)

E_2 = (k_e*q2)/(2a)^2 = (k_e*q2)/(4a^2)

Ex = (E_1 - E_2)*2/sqrt(2)

Ey = (E_1 + E_2)*2/sqrt(2)

E = 2/sqrt(2)*(E_1 - E_2) i + 2/sqrt(2)*(E_1 + E_2) j = 2/sqrt(2)*(k_e/(4a^2))*(q1 - q2) i + 2/sqrt(2)*(k_e/(4a^2))*(q1 + q2)

I am stuck here; I'm not sure if I've been going about this correctly or what steps to take next.

Thank you.

Hello JessieS , Welcome to PF !

Are you given any numerical values, particularly any for E_x and E_y ?

 

[JessieS]

Hello JessieS , Welcome to PF !

Are you given any numerical values, particularly any for E_x and E_y ?

Thank you!

And no I am not. I am supposed to just use Ex and Ey as variables.

 

[SammyS]

Thank you!

And no I am not. I am supposed to just use Ex and Ey as variables.

OK.

You are on the right track.

First, at least one error. What is the distance from q₁ to A and q₂ to A. The square of each of those distances is 2a², not 4a² .

I suggest that you keep E_x and E_y separate, rather than lumping them together into one big vector expression.

You have that E_x = C⋅(q₁ - q₂) and E_y = C⋅(q₁ + q₂) , where the coefficient, C is made up of all that stuff in your equation.

 

[JessieS]

OK.

You are on the right track.

First, at least one error. What is the distance from q₁ to A and q₂ to A. The square of each of those distances is 2a², not 4a² .

I suggest that you keep E_x and E_y separate, rather than lumping them together into one big vector expression.

You have that E_x = C⋅(q₁ - q₂) and E_y = C⋅(q₁ + q₂) , where the coefficient, C is made up of all that stuff in your equation.

Ok, so could I do this?

Since
E_x = 2/√2*(q₁ - q₂)*(k_e/(2a²))

E_y = 2/√2*(q₁ + q₂)*(k_e/(2a²))

Then
(q₁ - q₂) = (E_x * a²*√2)/k_e
+ (q₁ + q₂) = (E_y * a²*√2)/k_e
--------------------------------
q₁ = (a²*√2)/k_e * (E_x + E_y)

So then

q₂ = (a²*√2)/k_e * (E_y - E_x)

Is that correct?

 

[SammyS]

Ok, so could I do this?

Since
E_x = 2/√2*(q₁ - q₂)*(k_e/(2a²))

E_y = 2/√2*(q₁ + q₂)*(k_e/(2a²))

Then
(q₁ - q₂) = (E_x * a²*√2)/k_e
+ (q₁ + q₂) = (E_y * a²*√2)/k_e
--------------------------------
q₁ = (a²*√2)/k_e * (E_x + E_y)

So then

q₂ = (a²*√2)/k_e * (E_y - E_x)

Is that correct?

Yes. That's it.

Subtracting should give q₂.

 

[JessieS]

Yes. That's it.

Subtracting should give q₂.

Thank you for your help!