A -6.00 µC charge is moving with a speed of 6.60 104 m/s parallel to a very long, straight wire. The wire is 5.00 cm from the charge and carries a current of 73.0 A in a direction opposite to that of the moving charge. Find the magnitude and direction of the force on the charge.
To find the magnitude and direction of the force on the charge, we can use the formula for the magnetic force on a moving charge:
F = qVB
where F is the force, q is the charge, V is the velocity, and B is the magnetic field.
Step 1: Calculate the magnetic field created by the current-carrying wire at a distance of 5.00 cm.
The magnetic field produced by a current-carrying wire at a distance r is given by:
B = (μ0 * I) / (2π * r)
where μ0 is the permeability of free space, I is the current, and r is the distance.
Plugging in the values:
μ0 = 4π * 10^-7 Tm/A (permeability of free space)
I = -73.0 A (opposite direction to the moving charge)
r = 5.00 cm = 0.05 m (distance)
B = (4π * 10^-7 Tm/A * -73.0 A) / (2π * 0.05 m)
B = -5.824 T
Step 2: Calculate the force on the charge.
F = qVB
q = -6.00 µC = -6.00 × 10^-6 C (charge)
V = 6.60 × 10^4 m/s (velocity)
B = -5.824 T (magnetic field)
F = (-6.00 × 10^-6 C) * (6.60 × 10^4 m/s) * (-5.824 T)
F ≈ 2.68 N
The magnitude of the force on the charge is approximately 2.68 N.
Step 3: Determine the direction of the force.
The direction of the force can be determined using the right hand rule. If we align our fingers with the velocity (V) in the direction of the moving charge and curl our fingers towards the magnetic field (B), then the thumb will point in the direction of the force.
In this case, with the given values, the force will be directed downward, perpendicular to the velocity and magnetic field.
So, the direction of the force is downward.
Therefore, the magnitude of the force on the charge is approximately 2.68 N, directed downward.
To find the magnitude and direction of the force on the charge, we can use the formula for the magnetic force on a moving charge in a magnetic field:
F = qvBsinθ
where:
F is the force on the charge,
q is the charge of the moving particle,
v is the velocity of the moving charge,
B is the magnetic field,
θ is the angle between the velocity and the magnetic field.
First, let's calculate the magnetic field produced by the current-carrying wire at a distance of 5.00 cm. The formula to calculate the magnetic field due to a long straight wire is:
B = (μ0 * I) / (2π * r),
where:
B is the magnetic field,
μ0 is the permeability of free space (4π * 10^-7 T·m/A),
I is the current in the wire,
r is the distance from the wire.
Plugging in the given values, we have:
μ0 = 4π * 10^-7 T·m/A,
I = 73.0 A,
r = 5.00 cm = 0.05 m.
B = (4π * 10^-7 T·m/A * 73.0 A) / (2π * 0.05 m)
B = 2.34 × 10^-5 T
Now, let's calculate the angle θ between the velocity of the charge and the magnetic field. Since the wire carries a current in a direction opposite to that of the moving charge, the magnetic field will be directed opposite to the velocity. This means the angle between them is 180 degrees.
θ = 180 degrees = π radians
Finally, we can calculate the force on the charge:
F = qvBsinθ
F = (-6.00 × 10^-6 C) * (6.60 × 10^4 m/s) * (2.34 × 10^-5 T) * sin(π)
F ≈ -5.60 × 10^-1 N
The magnitude of the force is approximately 0.56 N, and the direction is negative, indicating that the force is directed opposite to the velocity of the charge.