A loop of surface area s1=10cm^2 placed ina uniform magnetic field Bof magnitude B1=8×10^-2T, has its plane perpendicular to the field lines . A second loop of area s2=15cm^2 is placed in another uniform field B2 of magnitude B2=0.1T such that its normal unit vector n makes an angle 2 theta 30º with lines of field B2.
1) which loop recieves more magnetic flux?why
2) Are there any induced currents in these two loops? Justify
To determine which loop receives more magnetic flux, we need to calculate the magnetic flux for each loop. The magnetic flux (Φ) through a loop is given by the formula Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field lines and the normal vector of the loop.
1) For the first loop with surface area s1 = 10 cm^2 and magnetic field magnitude B1 = 8 × 10^-2 T, we know that the loop is perpendicular to the magnetic field lines, so θ1 = 0°. The magnetic flux through the first loop is Φ1 = B1 * s1 = 8 × 10^-2 T * 10 cm^2 = 8 × 10^-1 T cm^2.
2) For the second loop with surface area s2 = 15 cm^2 and magnetic field magnitude B2 = 0.1 T, the angle between the field lines and the normal vector is 2θ = 30°. Therefore, the angle between the normal vector and the field lines is θ2 = 15°. The magnetic flux through the second loop is Φ2 = B2 * s2 * cos(θ2) = 0.1 T * 15 cm^2 * cos(15°).
Comparing the magnetic flux of the two loops:
Φ1 = 8 × 10^-1 T cm^2
Φ2 = 0.1 T * 15 cm^2 * cos(15°)
To determine which loop receives more magnetic flux, we need to calculate the values for Φ2. Substituting the values, we find:
Φ2 = 0.1 T * 15 cm^2 * cos(15°) ≈ 2.565 T cm^2
Since Φ2 > Φ1, the second loop receives more magnetic flux than the first loop.
Now, let's move on to the second question:
To determine if there are any induced currents in these two loops, we need to consider Faraday's law of electromagnetic induction. According to this law, an induced current is produced in a closed loop when there is a change in the magnetic flux passing through the loop.
In the case of the first loop, since its plane is perpendicular to the field lines, there will be no change in the magnetic flux. Therefore, no induced current will be produced in the first loop.
For the second loop, although its area is not changing, the angle between the normal vector and the field lines is changing as θ2 varies. This variation in the angle will cause a change in the magnetic flux passing through the loop, resulting in an induced current in the second loop.
In conclusion, there is no induced current in the first loop, while there is an induced current in the second loop due to the change in magnetic flux caused by the varying angle between the normal vector and the field lines.
To determine which loop receives more magnetic flux, we can use the formula for magnetic flux:
Φ = B * A * cos(θ),
where Φ is the magnetic flux, B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field lines and the normal to the loop.
1) To compare the magnetic flux received by the two loops, let's calculate the magnetic flux for each loop:
For the first loop:
Φ1 = B1 * s1 * cos(90°),
Φ1 = 8×10^-2 T * 10 cm^2 * cos(90°).
Since the first loop is perpendicular to the magnetic field lines (θ = 90°), the term cos(90°) becomes zero. Hence, Φ1 = 0.
For the second loop:
Φ2 = B2 * s2 * cos(2θ),
Φ2 = 0.1 T * 15 cm^2 * cos(2 * 30°).
Simplifying this equation, we have:
Φ2 = 0.1 T * 15 cm^2 * cos(60°),
Φ2 = 0.1 T * 15 cm^2 * 0.5,
Φ2 = 0.75 T cm^2.
Thus, the second loop receives more magnetic flux as its value is 0.75 T cm^2, while the first loop receives no magnetic flux (0).
2) To determine if there are any induced currents in these two loops, we need to apply Faraday's law of electromagnetic induction, which states that an induced electromotive force (EMF) is generated in a closed loop when the magnetic flux through the loop changes.
The induced EMF can be calculated using the equation:
ε = -dΦ/dt,
where ε is the induced EMF and dΦ/dt is the rate of change of magnetic flux.
For the first loop, since it receives no magnetic flux (0), there will be no change in magnetic flux, and therefore, no induced EMF or current.
For the second loop, if the magnetic field B2 changes with time, then there will be a change in magnetic flux and consequently an induced EMF and current. However, without knowing the temporal changes in the magnetic field B2, we cannot conclusively determine if there will be induced currents in the second loop.
To summarize, the second loop receives more magnetic flux than the first loop. As for induced currents, the first loop does not have any induced currents, while the presence of induced currents in the second loop depends on the specific temporal changes in the magnetic field B2.