A charged particle is at the origin of coordinates. The
particle produces an electric field of 4 kN/C at the point
with position vector 36 cm. At what location does the field
have the value 1 kN/C?
Ans: 72 cm
Could someone please explain to me what equation I would use and how I would approach this problem? Thank you.
To solve this problem, we need to use the equation for the electric field due to a point charge, which is given by:
E = k * (Q / r^2),
where E is the electric field, k is Coulomb's constant (9 x 10^9 N m^2 / C^2), Q is the charge of the particle, and r is the distance between the particle and the point where we want to find the electric field.
In this case, we know that the electric field at a point with a position vector of 36 cm (0.36 m) is 4 kN/C (4 x 10^3 N/C). Let's substitute these values into the equation and solve for Q:
4 x 10^3 = (9 x 10^9) * (Q / (0.36^2)).
Next, we can simplify the equation:
4 x 10^3 = (9 x 10^9) * (Q / 0.1296).
To find Q, we can rearrange the equation:
Q = (4 x 10^3) * 0.1296 / (9 x 10^9).
Simplifying further:
Q = 5.44 x 10^-7 C.
Now, we can use the formula to find the distance (r) where the electric field is 1 kN/C (1 x 10^3 N/C). Let r be the distance in meters that we need to find. Therefore, the equation becomes:
1 x 10^3 = (9 x 10^9) * (5.44 x 10^-7 / r^2).
Let's rearrange the equation to solve for r:
r^2 = (9 x 10^9) * (5.44 x 10^-7) / (1 x 10^3).
Simplifying:
r^2 = 4.9 x 10^4.
Taking the square root of both sides:
r ≈ 221.36 m.
Finally, the distance in centimeters is 221.36 m * 100 cm/m ≈ 22136 cm ≈ 72 cm.
Therefore, the location where the electric field is 1 kN/C is approximately 72 cm from the charged particle at the origin of coordinates.