A long straight horizontal wire carries a current I = 4.80 A to the left. A positive 1.00 C charge moves to the right at a distance 5.00 m above the wire at constant speed v = 4500 m/s . What is the magnitude B of the magnetic field at the location of the charge due to the current-carrying wire?
i tried B= (mu*I/2*pi*r). It keeps saying its wrong? help!
That's the right equation.
Got me.
To find the magnitude of the magnetic field at the location of the charge due to the current-carrying wire, you can use the Biot-Savart law. The formula you mentioned, B = (mu*I/2*pi*r), is a simplified version of the Biot-Savart law equation for an infinitely long wire. However, in this case, the wire is not infinitely long, and we need to take into account the distance between the wire and the charge.
Here's the step-by-step solution:
1. First, let's identify the given information:
- Current in the wire: I = 4.80 A (to the left)
- Charge's velocity: v = 4500 m/s (to the right)
- Distance from the wire to the charge: d = 5.00 m
2. To find the magnetic field B, we can use the Biot-Savart law equation:
B = (mu0 * I) / (2 * pi * r),
where mu0 (mu-zero) is the permeability of free space, equal to 4π x 10^(-7) T·m/A.
3. Now, let's calculate the value of r. Since the charge is moving directly above the wire, the distance r is equal to the horizontal distance between the wire and the charge. We can use Pythagoras' theorem to calculate r:
r = sqrt(d^2 + v^2 * t^2),
where t is the time the charge takes to pass over the wire. Since it's moving at a constant speed, we can calculate the time using the formula:
t = d / v.
Substituting the values into the equations:
r = sqrt((5.00 m)^2 + (4500 m/s)^2 * (5.00 m / 4500 m/s)^2),
4. Solve the equation for r:
r = sqrt(25.00 m^2 + (1.00 m)^2) = sqrt(26.00 m^2) = 5.10 m.
5. Now we can substitute the values of mu0, I, and r into the Biot-Savart law equation to find B:
B = (4π x 10^(-7) T·m/A * 4.80 A) / (2π * 5.10 m)
= (4π x 10^(-7) * 4.80 A) / (2π * 5.10 m)
= (1.92 x 10^(-6) T·m) / (10.20 m)
≈ 1.88 x 10^(-7) T,
where T represents tesla, the unit of magnetic field.
Therefore, the magnitude of the magnetic field at the location of the charge is approximately 1.88 x 10^(-7) Tesla.
To find the magnitude of the magnetic field at the location of the charge due to the current-carrying wire, you can use the Biot-Savart law.
The formula you mentioned, B = (μ * I) / (2 * π * r), is correct. However, there seems to be an issue with the parameters you provided.
In the Biot-Savart law formula, μ represents the permeability of free space, which is a constant equal to 4π × 10^(-7) T·m/A.
I represents the current flowing through the wire, which you correctly stated as I = 4.80 A.
The r in the formula stands for the distance between the wire and the charge. However, in your question, you provided the distance of 5.00 m above the wire. To use this value in the formula, you need to consider the minimum distance between the wire and the charge, which is the perpendicular distance. In this case, the perpendicular distance would be the same as the distance stated: 5.00 m.
So, substituting the values into the formula, we have:
B = (μ * I) / (2 * π * r)
B = (4 * π * 10^(-7) T·m/A * 4.80 A) / (2 * π * 5.00 m)
By simplifying and canceling out the π, we get:
B = (2 * 10^(-7) T·m/A * 4.80 A) / 5.00 m
B = (9.60 * 10^(-7) T·m) / 5.00 m
The meters (m) in both the numerator and denominator cancel out, leaving us with the final result:
B = 1.92 * 10^(-7) T
Therefore, the magnitude of the magnetic field at the location of the charge due to the current-carrying wire is 1.92 * 10^(-7) T.