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.
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 force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
The formula for Coulomb's Law is:
F = k * (q1 * q2) / r^2
Where F is the force, k is the electrostatic constant (k = 8.99 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
In this case, the charge of the wire is distributed along its length, so to calculate the force on the wire, we need to integrate the force contributed by each infinitesimally small segment of the wire.
The linear charge density (λ) is given as 7.50 x 10^-4 C/m, which means that in every meter of wire, there is a charge of 7.50 x 10^-4 C.
First, we need to calculate the total charge (Q) in a small segment of the wire. We can do this by multiplying the linear charge density (λ) by the length of the small segment (Δl):
ΔQ = λ * Δl
Then, we can calculate the force (ΔF) between the point charge q and the small segment of the wire using Coulomb's Law:
ΔF = k * (q * ΔQ) / r^2
Finally, we integrate the force (ΔF) over the entire length of the wire to find the total force (F).
Let's calculate the magnitude of the force exerted by the charge q on the wire step-by-step:
Step 1: Calculate the length of the small segment (Δl)
No information given, assuming we are dealing with infinitesimally small segment.
Step 2: Calculate the total charge (ΔQ) in the small segment
ΔQ = λ * Δl = (7.50 x 10^-4 C/m) * (Δl)
Step 3: Calculate the force (ΔF) between the point charge q and the small segment of the wire
ΔF = k * (q * ΔQ) / r^2 = (8.99 x 10^9 N m^2/C^2) * (15.0 x 10^-6 C) * (7.50 x 10^-4 C/m) * (Δl) / (0.045 m)^2
Step 4: Integrate the force (ΔF) over the entire length of the wire to find the total force (F)
F = ∫ ΔF
Unfortunately, without the information about the length of the wire or the limits of integration, it is not possible to evaluate the definite integral or calculate the total force.
To find the magnitude of the force exerted by the charge q on the wire, we can use Coulomb's law:
F = k * (|q1| * |q2|) / r^2
where F is the force, k is Coulomb's constant (9.0 × 10^9 N·m²/C²), q1 is the charge on the wire, q2 is the charge q, and r is the distance between the charges.
First, let's convert the charge density of the wire to an actual charge. The linear charge density (λ) is given by:
λ = Q / L
Where Q is the total charge and L is the length of the wire. Rearranging the equation, we can solve for Q:
Q = λ * L
In this case, the linear charge density is given as 7.50 × 10^-4 C/m. However, we do not know the length of the wire (L). Without that information, we cannot accurately calculate the total charge on the wire.
Therefore, it is not possible to directly calculate the magnitude of the force exerted by the charge q on the wire without knowing the length of the wire.