An electron is located at the origin. A proton is located at (2, 0). Find the electric field direction at (1, 1).
because the charges are opposite you have some symettry here: the E in the x direction is zero, in the y direction it is E=kq/2 *(2)=kq
To find the electric field direction at a point, you can use Coulomb's law. Coulomb's law states that the electric field created by a charged particle is directly proportional to the charge and inversely proportional to the square of the distance between the two points.
The formula for the electric field created by a point charge is given by:
E = k * q / r²
Where:
- E is the electric field
- k is the electrostatic constant (k = 8.99 × 10^9 N m²/C²)
- q is the charge of the particle creating the electric field
- r is the distance between the point charge and the point at which the electric field is being measured
In this case, the electron at the origin has a charge of -1.6 × 10^-19 C (Coulombs), and the proton at (2, 0) has a charge of +1.6 × 10^-19 C.
To find the electric field at point (1, 1), we need to calculate the electric field due to the electron and the electric field due to the proton and then combine them vectorially.
1. Electric field due to the electron:
The distance between the electron at the origin and the point (1, 1) is given by:
r_electron = sqrt((x2 - x1)² + (y2 - y1)²)
= sqrt((0 - 1)² + (0 - 1)²)
= sqrt(2)
Plugging in the values into Coulomb's law, we get:
E_electron = k * q_electron / r_electron²
= (8.99 × 10^9) * (-1.6 × 10^-19) / (sqrt(2))²
= -2.546 × 10^9 N/C
The electric field due to the electron is in the negative x and y directions.
2. Electric field due to the proton:
The distance between the proton at (2, 0) and the point (1, 1) is given by:
r_proton = sqrt((x2 - x1)² + (y2 - y1)²)
= sqrt((2 - 1)² + (0 - 1)²)
= sqrt(2)
Using Coulomb's law with the proton's charge, we get:
E_proton = k * q_proton / r_proton²
= (8.99 × 10^9) * (1.6 × 10^-19) / (sqrt(2))²
= 2.546 × 10^9 N/C
The electric field due to the proton is in the positive x direction and negative y direction.
To find the net electric field, we need to add the electric field vectors due to the electron and the proton:
E_total = E_electron + E_proton
Substituting the values, we get:
E_total = (-2.546 × 10^9) i - (2.546 × 10^9) j
Therefore, the electric field direction at point (1, 1) is (-2.546 × 10^9) i - (2.546 × 10^9) j. This means that it points towards the negative x direction and the positive y direction.