A long, straight wire has a uniform linear charge density of 7.50X10^-4 C/m. A point charge q=15.0 microcoulombs is located 4.50 cm away from the wire. Find the magnitude of the force exerted by the charge q on the wire.
You will have to perform a Coulomb's- Law integration of the force due to individual line elements of length dx, each with charge 7.50X10^-4 dx.
The answer will depend upon which part of the wire the point charge is "below" . They probably expect you to assume that it is below the middle. That will make the integration easier since forces along the wire direction will cancel due to symmetry.
To find the magnitude of the force exerted by the charge q on the wire, we can use Coulomb's law.
Coulomb's law states that the magnitude of the electrostatic force F between two point charges q1 and q2 separated by a distance r is given by:
F = (k * |q1 * q2|) / r^2
where k is Coulomb's constant, |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
In this case, we have a point charge q = 15.0 microcoulombs (μC), and it is located 4.50 cm away from the wire, which has a uniform linear charge density of λ = 7.50 × 10^-4 C/m.
To proceed, we need to convert the charge density from C/m to C/cm. We can do this by multiplying λ by 100:
λ = 7.50 × 10^-4 C/m = 7.50 × 10^-2 C/cm
Now, we can calculate the magnitude of the force exerted by the charge q on the wire using Coulomb's law.
Step 1: Convert the distance from cm to meters:
r = 4.50 cm = 4.50 × 10^-2 m
Step 2: Substitute the values into the formula:
F = (k * |q * λ|) / r^2
Step 3: Calculate the force:
F = (9.0 × 10^9 Nm^2/C^2 * |15.0 × 10^-6 C * 7.50 × 10^-2 C/cm|) / (4.50 × 10^-2 m)^2
F = (9.0 × 10^9 Nm^2/C^2 * 1.125 × 10^-6 C^2) / (2.025 × 10^-3 m^2)
F ≈ 4.84 N (rounded to two significant figures)
Therefore, the magnitude of the force exerted by the charge q on the wire is approximately 4.84 Newtons.