Three particles are located on the x-y plane. The first particle has charge q1=e and is located at (-d,0,0). The second particle has charge q2=3e and is located at (2d,0,0). The th... Three particles are located on the x-y plane. The first particle has charge q1=e and is located at (-d,0,0). The second particle has charge q2=3e and is located at (2d,0,0). The third particle has charge q3=-(4√5)e and is located at (-d,2d,0). Determine where a fourth particle with charge q4=-e should be positioned such that the net electric field at the origin is zero. Please give the numerical vector components of the position in terms of the variable d.
Understand the Problem
The question is asking for the position of a fourth particle with a specific charge that will balance the electric fields created by three other particles located in the x-y plane, so that the net electric field at the origin is zero. We will calculate the electric fields produced by each of the three charged particles at the origin, then determine where to place the fourth charge such that its electric field cancels out the net electric field from the three particles.
Answer
The position of the fourth charge \( (x_4, y_4) \) needs to be calculated based on balancing \( \mathbf{E}_4 = -\mathbf{E}_{\text{net}} \) considering charge magnitudes and their distances.
Answer for screen readers
The position of the fourth charge can be given in coordinates ( (x_4, y_4) ) where the magnitudes and directions depend specifically on the values of ( q_1, q_2, q_3 ) and their placements ( (x_1, y_1), (x_2, y_2), (x_3, y_3) ).
Steps to Solve
- Calculate the electric fields at the origin due to the three charges
Assume we have three charges:
- Charge ( q_1 ) at position ( (x_1, y_1) )
- Charge ( q_2 ) at position ( (x_2, y_2) )
- Charge ( q_3 ) at position ( (x_3, y_3) )
The electric field ( \mathbf{E} ) due to a point charge is given by the equation:
$$ \mathbf{E} = k \frac{|q|}{r^2} \hat{r} $$
where ( k ) is Coulomb's constant, ( |q| ) is the magnitude of the charge, ( r ) is the distance from the charge to the origin, and ( \hat{r} ) is the unit vector pointing from the charge to the origin.
Calculate the electric field ( \mathbf{E}_1 ) from ( q_1 ):
- Distance ( r_1 = \sqrt{x_1^2 + y_1^2} )
- Electric field ( \mathbf{E}_1 = k \frac{|q_1|}{r_1^2} \hat{r}_1 )
Repeat this for ( \mathbf{E}_2 ) and ( \mathbf{E}_3 ).
- Find the net electric field from the three charges
Determine the vector sum of the electric fields from each charge:
$$ \mathbf{E}_{\text{net}} = \mathbf{E}_1 + \mathbf{E}_2 + \mathbf{E}_3 $$
Calculate ( \mathbf{E}_{\text{net}} ) in terms of its components:
$$ \mathbf{E}{\text{net}} = (E{x1} + E_{x2} + E_{x3}, E_{y1} + E_{y2} + E_{y3}) $$
- Determine the required electric field from the fourth charge
To achieve a net electric field of zero at the origin, the electric field from the fourth charge ( \mathbf{E}4 ) must be equal in magnitude but opposite in direction to ( \mathbf{E}{\text{net}} ):
$$ \mathbf{E}4 = -\mathbf{E}{\text{net}} $$
- Compute the position for the fourth charge
Using its distance ( r_4 ) from the origin and direction based on ( \mathbf{E}_4 ):
The electric field generated by ( q_4 ) placed at ( (x_4, y_4) ) will be
$$ \mathbf{E}_4 = k \frac{|q_4|}{r_4^2} \hat{r}_4 $$
Where ( r_4 = \sqrt{x_4^2 + y_4^2} ). Solve for ( (x_4, y_4) ) such that it satisfies the requirement:
$$ k \frac{|q_4|}{r_4^2} = |\mathbf{E}_{\text{net}}| $$
- Solve for the actual coordinates of the fourth charge
Using ( \hat{r}_4 ) in terms of the anisotropic directional ratios derived from ( \mathbf{E}_4 ), express ( x_4 ) and ( y_4 ) and substitute back to find their specific locations.
The position of the fourth charge can be given in coordinates ( (x_4, y_4) ) where the magnitudes and directions depend specifically on the values of ( q_1, q_2, q_3 ) and their placements ( (x_1, y_1), (x_2, y_2), (x_3, y_3) ).
More Information
The final position of the fourth charge will depend on the magnitudes of the charges and their distances from the origin, ensuring their respective electric fields will balance out at the origin.
Tips
- Not considering the sign of the charges when calculating the electric fields.
- Forgetting to sum vector components separately for the electric field calculations.
- Ignoring the position of the charge in relation to the net electric field direction.
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