Question

Step 1, when solving a two dimensional, multi-charge problem, is to define the vectors. Please identify the next five steps, in order.

Answer 1

When solving a two-dimensional multi-charge problem, after defining the vectors, you can follow these steps:

### Step 2: Calculate the Electric Field from Each Charge
- Use Coulomb's Law to calculate the electric field (\( \vec{E} \)) produced by each charge at the location of interest. The electric field due to a point charge \( q \) at a distance \( r \) is given by:
\[
\vec{E} = \frac{k |q|}{r^2} \hat{r}
\]
where \( k \) is Coulomb's constant, and \( \hat{r} \) is a unit vector pointing away from the charge (or towards the charge, depending on whether \( q \) is positive or negative).

### Step 3: Resolve the Electric Fields into Components
- Break down the electric field vectors into their x and y components. For each charge, if \( \vec{E} \) is the electric field vector, you can find the components as follows:
\[
E_x = E \cos(\theta) \quad \text{and} \quad E_y = E \sin(\theta)
\]
where \( \theta \) is the angle between the electric field vector and the x-axis.

### Step 4: Sum the Electric Field Components
- Sum the x components and y components of the electric fields from all charges separately:
\[
E_{total\_x} = \sum E_{x\_i} \quad \text{and} \quad E_{total\_y} = \sum E_{y\_i}
\]

### Step 5: Calculate the Magnitude of the Total Electric Field
- Determine the magnitude of the total electric field vector using the Pythagorean theorem:
\[
E_{total} = \sqrt{(E_{total\_x})^2 + (E_{total\_y})^2}
\]

### Step 6: Determine the Direction of the Total Electric Field
- Find the direction (angle) of the total electric field relative to the x-axis:
\[
\theta_{total} = \tan^{-1}\left(\frac{E_{total\_y}}{E_{total\_x}}\right)
\]

### Step 7: Analyze the Results
- Interpret the results considering the context of the problem, such as the effect of the total electric field on a charged particle, or how the field would influence the movement of the charges in the system.

Following these steps will help you systematically analyze and solve multi-charge problems in two dimensions.