A 13.9 nC charge is located at position (x,y) = (1cm,2cm).
A. At what (x,y) position is the electrical field -225,000i_hat N/C?
B. At what (x,y) position is the electrical field (-161,000i_hat + 80,500j_hat) N/C?
C. At what (x,y) position is the electrical field (-21,600i_hat + 28,800j_hat) N/C?
To find the position at which a given electric field is located, we can use the formula for the electric field generated by a point charge. The formula is:
E = k * (Q / r^2)
Where:
- E is the electric field vector.
- k is the Coulomb's constant (k = 8.99 x 10^9 N m^2/C^2).
- Q is the charge generating the electric field.
- r is the distance between the point charge and the position where the electric field is being calculated.
We can rewrite the formula to solve for the distance r:
r^2 = k * (Q / E)
A. For an electric field of -225,000i_hat N/C:
Given:
- E = -225,000i_hat N/C
- Q = 13.9 x 10^-9 C
- k = 8.99 x 10^9 N m^2/C^2
Plugging these values into the equation, we have:
r^2 = (8.99 x 10^9 N m^2/C^2) * (13.9 x 10^-9 C) / (-225,000i_hat N/C)
Simplifying, we get:
r^2 = 0.554
Taking the square root of both sides, we find:
r ≈ 0.744 cm
Therefore, the position (x, y) at which the electric field is -225,000i_hat N/C is approximately (0.744 cm, 0 cm).
B. For an electric field of (-161,000i_hat + 80,500j_hat) N/C:
Given:
- E = (-161,000i_hat + 80,500j_hat) N/C
- Q = 13.9 x 10^-9 C
- k = 8.99 x 10^9 N m^2/C^2
To solve for the position (x, y), we need to find the distances in the x-direction and y-direction separately.
For the x-direction, we can use the formula:
r_x^2 = (8.99 x 10^9 N m^2/C^2) * (13.9 x 10^-9 C) / (-161,000 N/C)
Simplifying, we get:
r_x^2 = 0.781
Taking the square root, we have:
r_x ≈ 0.884 cm
For the y-direction, we can use the formula:
r_y^2 = (8.99 x 10^9 N m^2/C^2) * (13.9 x 10^-9 C) / (80,500 N/C)
Simplifying, we get:
r_y^2 = 0.227
Taking the square root, we have:
r_y ≈ 0.476 cm
Therefore, the position (x, y) at which the electric field is (-161,000i_hat + 80,500j_hat) N/C is approximately (0.884 cm, 0.476 cm).
C. For an electric field of (-21,600i_hat + 28,800j_hat) N/C:
Given:
- E = (-21,600i_hat + 28,800j_hat) N/C
- Q = 13.9 x 10^-9 C
- k = 8.99 x 10^9 N m^2/C^2
Using the same approach as before, we find:
For the x-direction:
r_x^2 = (8.99 x 10^9 N m^2/C^2) * (13.9 x 10^-9 C) / (-21,600 N/C)
r_x ≈ 0.872 cm
For the y-direction:
r_y^2 = (8.99 x 10^9 N m^2/C^2) * (13.9 x 10^-9 C) / (28,800 N/C)
r_y ≈ 0.685 cm
Therefore, the position (x, y) at which the electric field is (-21,600i_hat + 28,800j_hat) N/C is approximately (0.872 cm, 0.685 cm).
To answer these questions, we need to use Coulomb's law and the principle of superposition.
Coulomb's law states that the electric field (E) created by a point charge (Q) at a distance (r) from the charge is given by:
E = k * Q / r^2
where k is Coulomb's constant (k ≈ 9 × 10^9 N•m^2/C^2).
The electric field is a vector quantity, meaning it has both direction and magnitude. In two-dimensional problems like this, we can represent the electric field as a vector in Cartesian coordinates using i-hat (unit vector in x-direction) and j-hat (unit vector in y-direction).
A. To find the position (x,y) where the electric field is -225,000i_hat N/C:
Given:
Charge Q = 13.9 nC = 13.9 × 10^-9 C
Electric field E = -225,000i_hat N/C
Distance r between the charge and the point where the electric field is measured is unknown.
Using Coulomb's law, we can rearrange the equation to solve for the distance r:
r^2 = k * Q / |E|
Substituting the known values:
r^2 = (9 × 10^9 N•m^2/C^2) * (13.9 × 10^-9 C) / (225,000 N/C)
r^2 ≈ 5.27 × 10^-4 m^2
Therefore, r ≈ √(5.27 × 10^-4) ≈ 0.023 m ≈ 2.3 cm
So, the (x,y) position where the electric field is -225,000i_hat N/C is approximately (1 cm ± 2.3 cm, 2 cm).
B. To find the position (x,y) where the electric field is (-161,000i_hat + 80,500j_hat) N/C:
Given:
Charge Q = 13.9 nC = 13.9 × 10^-9 C
Electric field E = (-161,000i_hat + 80,500j_hat) N/C
Distance r between the charge and the point where the electric field is measured is unknown.
Similarly, we can use Coulomb's law to find r:
r^2 = k * Q / |E|
Substituting the known values:
r^2 = (9 × 10^9 N•m^2/C^2) * (13.9 × 10^-9 C) / [(161,000^2 + 80,500^2) N^2/C^2]
r^2 ≈ 2.90 × 10^-4 m^2
Therefore, r ≈ √(2.90 × 10^-4) ≈ 0.017 m ≈ 1.7 cm
So, the (x,y) position where the electric field is (-161,000i_hat + 80,500j_hat) N/C is approximately (1 cm ± 1.7 cm, 2 cm ± 1.7 cm).
C. To find the position (x,y) where the electric field is (-21,600i_hat + 28,800j_hat) N/C:
Given:
Charge Q = 13.9 nC = 13.9 × 10^-9 C
Electric field E = (-21,600i_hat + 28,800j_hat) N/C
Distance r between the charge and the point where the electric field is measured is unknown.
Using Coulomb's law, we can again find r:
r^2 = k * Q / |E|
Substituting the known values:
r^2 = (9 × 10^9 N•m^2/C^2) * (13.9 × 10^-9 C) / [(21,600^2 + 28,800^2) N^2/C^2]
r^2 ≈ 1.92 × 10^-4 m^2
Therefore, r ≈ √(1.92 × 10^-4) ≈ 0.014 m ≈ 1.4 cm
So, the (x,y) position where the electric field is (-21,600i_hat + 28,800j_hat) N/C is approximately (1 cm ± 1.4 cm, 2 cm ± 1.4 cm).
Note: The ± sign indicates that the position can be either slightly to the left or right (in x-direction) and slightly above or below (in y-direction) the given point (1 cm, 2 cm).