A 5.30-ìC charge is moving with a speed of 9.30 x 104 m/s parallel to a very long, straight wire. The wire is 6.60 cm from the charge and carries a current of 87.0 A. Find the magnitude of the force on the charge.
The magnetic field at the charge location is
B = mu_o* I/(2*pi*R)
where mu_o is the permeability of free space, 4*pi*10^-7 Weber/(amp*meter)
The force on the charge is
q V B
since V and B are in perpendicular directions
To find the magnitude of the force on the charge, we can use the formula for the magnetic force experienced by a moving charge in a magnetic field.
The formula for the magnetic force (F) on a moving charge in a magnetic field is given by:
F = q * v * B * sin(θ)
Where:
- F is the magnitude of the force
- q is the charge of the particle
- v is the velocity of the particle
- B is the magnetic field strength
- θ is the angle between the velocity vector and the magnetic field vector
In this case, we are given:
- q = 5.30 x 10^(-6) C (charge of the particle)
- v = 9.30 x 10^4 m/s (velocity of the particle)
- B (magnetic field strength) = ? (needs to be calculated)
- θ (angle between the velocity and the magnetic field) = 90° (since the wire is perpendicular to the charge's velocity)
Now, let's calculate the magnetic field strength (B) using the formula:
B = μ₀ * I / (2πd)
Where:
- μ₀ is the permeability of free space, approximately 4π x 10^(-7) T·m/A
- I is the current flowing through the wire
- d is the distance between the wire and the charge
In this case, we are given:
- I = 87.0 A (current flowing through the wire)
- d = 6.60 cm = 6.60 x 10^(-2) m (distance between the wire and the charge)
Now, we can substitute the given values into the formula to find the magnetic field strength (B).
B = (4π x 10^(-7) T·m/A) * (87.0 A) / (2π * 6.60 x 10^(-2) m)
After simplifying the expression, we get:
B ≈ 5.27 x 10^(-6) T (Tesla)
Now we have all the values we need to calculate the magnitude of the force (F). Substituting the values into the formula:
F = (5.30 x 10^(-6) C) * (9.30 x 10^4 m/s) * (5.27 x 10^(-6) T) * sin(90°)
Since sin(90°) = 1, we can simplify further:
F ≈ 5.30 x 10^(-6) C * 9.30 x 10^4 m/s * 5.27 x 10^(-6) T
After performing the multiplication, we get:
F ≈ 2.43 x 10^(-3) N
Therefore, the magnitude of the force on the charge is approximately 2.43 x 10^(-3) Newtons.