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Suppose we have two point charges in the Cartesian coordinate system. $q_1= e, q_2 = 2e$, where $q_1$ is positioned at $(0,0,0)$ and $q_2$ at $(a,0,0)$ for $a > 0 $. Further there is a point charge $q_3 = -e $ starting at infinty and I wanna now how much work does it need to come to the point $A = (0,a,0)$. I have two different ideas how to approach this:

1. I know oppositely charged objects attract each other so it will just "fly" there without any help, more I can say it has a higher potential at infinty than at the point A

2. I know this equation for the potential of a charge in a system of charges $Q_1,\dots,Q_N$ $$\phi(\vec{x}) = \frac{1}{4\pi \epsilon_0} \sum_{i=1}^N \frac{Q_i}{\|\vec{x}-\vec{x}_i\|} $$ Considering $$\phi(\vec{x}) = \int_{-\infty}^{\vec{x}} \vec{E(\vec{x})} d\vec{x}$$ these two equations I should get the desired result. (I wonder why this equation doesn't depend on my test charge.)

My question is which approach is the correct one? Where are my mistakes?

This is my first physics related question on English so please tell me if there are obscurities.

Suppose we have two point charges in the Cartesian coordinate system. $q_1= e, q_2 = 2e$, where $q_1$ is positioned at $(0,0,0)$ and $q_2$ at $(a,0,0)$ for $a > 0 $. Further there is a point charge $q_3 = -e $ starting at infinty and I wanna now how much work does it need to come to the point $A = (0,a,0)$. I have two different ideas how to approach this:

1. I know oppositely charged objects attract each other so it will just "fly" there without any help, more I can say it has a higher potential at infinty than at the point A

2. I know this equation for the potential of a charge in a system of charges $Q_1,\dots,Q_N$ $$\phi(\vec{x}) = \frac{1}{4\pi \epsilon_0} \sum_{i=1}^N \frac{Q_i}{\|\vec{x}-\vec{x}_i\|} $$ Considering $$\phi(\vec{x}) = \int_{-\infty}^{\vec{x}} \vec{E}(\vec{x}) d\vec{x}$$ these two equations I should get the desired result. (I wonder why this equation doesn't depend on my test charge.)

My question is which approach is the correct one? Where are my mistakes?

EDIT: As pointed out in the comments i calculate the potential: $$\phi(\vec{x}) = \frac{e}{4\pi \epsilon_0 a} (1+\sqrt{2})$$ I still wonder why approach one isn't true and why the potential doesnt depend on my test charge, though i think i can read the second in a book and will try to do so.

This is my first physics related question on English so please tell me if there are obscurities.

Suppose we have two point charges in the Cartesian coordinate system. $q_1= e, q_2 = 2e$, where $q_1$ is positioned at $(0,0,0)$ and $q_2$ at $(a,0,0)$ for $a > 0 $. Further there is a point charge $q_3 = -e $ starting at infinty and I wanna now how much energy does it need to come to the point $A = (0,a,0)$. I have two different ideas how to approach this:

1. I know oppositely charged objects attract each other so it will just "fly" there without any help, more I can say it has a higher potential at infinty than at the point A

2. I know this equation for the potential of a charge in a system of charges $Q_1,\dots,Q_N$ $$\phi(\vec{x}) = \frac{1}{4\pi \epsilon_0} \sum_{i=1}^N \frac{Q_i}{\|\vec{x}-\vec{x}_i\|} $$ Considering $$\phi(\vec{x}) = \int_{-\infty}^{\vec{x}} \vec{E(\vec{x})} d\vec{x}$$ these two equations I should get the desired result. (I wonder why this equation doesn't depend on my test charge.)

My question is which approach is the correct one? Where are my mistakes?

This is my first physics related question on English so please tell me if there are obscurities.

Suppose we have two point charges in the Cartesian coordinate system. $q_1= e, q_2 = 2e$, where $q_1$ is positioned at $(0,0,0)$ and $q_2$ at $(a,0,0)$ for $a > 0 $. Further there is a point charge $q_3 = -e $ starting at infinty and I wanna now how much work does it need to come to the point $A = (0,a,0)$. I have two different ideas how to approach this:

1. I know oppositely charged objects attract each other so it will just "fly" there without any help, more I can say it has a higher potential at infinty than at the point A

2. I know this equation for the potential of a charge in a system of charges $Q_1,\dots,Q_N$ $$\phi(\vec{x}) = \frac{1}{4\pi \epsilon_0} \sum_{i=1}^N \frac{Q_i}{\|\vec{x}-\vec{x}_i\|} $$ Considering $$\phi(\vec{x}) = \int_{-\infty}^{\vec{x}} \vec{E(\vec{x})} d\vec{x}$$ these two equations I should get the desired result. (I wonder why this equation doesn't depend on my test charge.)

My question is which approach is the correct one? Where are my mistakes?

This is my first physics related question on English so please tell me if there are obscurities.

Suppose we have two point charges in the cartesian coordinate system. $q_1= e, q_2 = 2e$, where $q_1$ is positioned at $(0,0,0)$ and $q_2$ at $(a,0,0)$ for $a > 0 $. Further there is a point charge $q_3 = -e $ starting at infinty and i wanna now how much energy does it need to come to the point $A = (0,a,0)$. I have two different ideas how to approach this:

1. i know oppositley charged objects attract each other so it will just "fly" there without any help, more i can say it has a higher potential at infinty than at the point A

2. I know this equation for the potential of a charge in a system of charges $Q_1,\dots,Q_N$ $$\phi(\vec{x}) = \frac{1}{4\pi \epsilon_0} \sum_{i=1}^N \frac{Q_i}{\|\vec{x}-\vec{x}_i\|} $$ Considering $$\phi(\vec{x}) = \int_{-\infty}^{\vec{x}} \vec{E(\vec{x})} d\vec{x}$$ these two equations i should get the desired result. (I wonder why this equation doesn't depend on my test charge.)

My question is which approach is the correct one  ? Where are my mistakes?

This is my first physics related question on english so please tell me if there are obscurities.

Suppose we have two point charges in the Cartesian coordinate system. $q_1= e, q_2 = 2e$, where $q_1$ is positioned at $(0,0,0)$ and $q_2$ at $(a,0,0)$ for $a > 0 $. Further there is a point charge $q_3 = -e $ starting at infinty and I wanna now how much energy does it need to come to the point $A = (0,a,0)$. I have two different ideas how to approach this:

1. I know oppositely charged objects attract each other so it will just "fly" there without any help, more I can say it has a higher potential at infinty than at the point A

2. I know this equation for the potential of a charge in a system of charges $Q_1,\dots,Q_N$ $$\phi(\vec{x}) = \frac{1}{4\pi \epsilon_0} \sum_{i=1}^N \frac{Q_i}{\|\vec{x}-\vec{x}_i\|} $$ Considering $$\phi(\vec{x}) = \int_{-\infty}^{\vec{x}} \vec{E(\vec{x})} d\vec{x}$$ these two equations I should get the desired result. (I wonder why this equation doesn't depend on my test charge.)

My question is which approach is the correct one? Where are my mistakes?

This is my first physics related question on English so please tell me if there are obscurities.