Considerable scientific work is currently underway to determine whether weak oscillating magnetic fields such as those found near outdoor power lines can affect human health. One study indicated that a magnetic field of magnitude 1.0 10-3 T, oscillating at 60 Hz, might stimluate red blood cells to become cancerous. If the diameter of a red blood cell is 8.5 µm, determine the maximum emf that can be generated around the perimeter of the cell if the magnetic field strength is 1.3 10-3 T.
To determine the maximum emf that can be generated around the perimeter of the red blood cell, we can use Faraday's law of electromagnetic induction.
Faraday's law states that the induced emf (ε) is equal to the rate of change of magnetic flux (Φ) through a loop of wire:
ε = -dΦ/dt
In this case, the magnetic field is oscillating at 60 Hz, so we need to find the maximum rate of change of magnetic flux.
The magnetic flux (Φ) through a loop can be calculated by multiplying the magnetic field strength (B) by the area (A) of the loop.
Φ = B * A
The area of the loop can be approximated as a circle, where the diameter of the red blood cell (d) is given as 8.5 µm. The radius (r) of the circle is half the diameter.
r = d/2
r = 8.5 µm / 2
r = 4.25 µm = 4.25 x 10^-6 m
The area (A) of the circle can be calculated using the formula:
A = π * r^2
A = π * (4.25 x 10^-6 m)^2
A = 5.67 x 10^-11 m^2
Now we have the magnetic field strength (B) as 1.3 x 10^-3 T and the area (A) as 5.67 x 10^-11 m^2. We can calculate the maximum magnetic flux (Φ) through the loop.
Φ = B * A
Φ = (1.3 x 10^-3 T) * (5.67 x 10^-11 m^2)
Φ = 7.371 x 10^-14 Wb (webers)
Next, we calculate the rate of change of magnetic flux by dividing the maximum magnetic flux (Φ) by the period (T) of the oscillating magnetic field, which is the inverse of the frequency (f).
T = 1/f
T = 1/60 Hz
T = 0.0167 s
Rate of change of magnetic flux (dΦ/dt) = Φ / T
Rate of change of magnetic flux (dΦ/dt) = 7.371 x 10^-14 Wb / 0.0167 s
Rate of change of magnetic flux (dΦ/dt) = 4.41 x 10^-12 Wb/s (weber per second)
Finally, we have the rate of change of magnetic flux (dΦ/dt). Since the induced emf (ε) is equal to the negative rate of change of magnetic flux, the maximum emf that can be generated around the perimeter of the red blood cell is:
ε = -dΦ/dt
ε = -(4.41 x 10^-12 Wb/s)
ε ≈ -4.41 x 10^-12 V (volts)
Therefore, the maximum emf that can be generated around the perimeter of the cell is approximately 4.41 x 10^-12 volts.
To determine the maximum emf (electromotive force) that can be generated around the perimeter of the red blood cell, we need to use Faraday's law of electromagnetic induction.
Faraday's law states that the induced emf in a closed loop is equal to the rate of change of magnetic flux through the loop. In this case, the magnetic field is oscillating at 60 Hz, which means it is changing with time.
The magnetic flux passing through a surface is given by the product of the magnetic field strength (B) and the enclosed area (A). However, since we are dealing with a circular loop (the perimeter of the red blood cell), the enclosed area will be the area of a circle.
The formula for the magnetic flux passing through a circular loop is:
Φ = B * A
The area of a circle can be calculated using the formula:
A = π * r^2
Given:
Magnetic field strength (B) = 1.3 * 10^-3 T
Diameter of the red blood cell = 8.5 µm = 8.5 * 10^-6 m
Radius of the red blood cell (r) = diameter / 2 = 8.5 * 10^-6 m / 2 = 4.25 * 10^-6 m
Plugging these values into the area formula:
A = π * (4.25 * 10^-6 m)^2 = π * (1.805625 * 10^-11 m^2)
Now, we can calculate the magnetic flux (Φ):
Φ = (1.3 * 10^-3 T) * (π * 1.805625 * 10^-11 m^2)
To find the maximum emf (ε), we can differentiate the magnetic flux with respect to time (t):
ε = -dΦ/dt
Since the magnetic field (B) is oscillating at 60 Hz, its angular frequency (ω) can be calculated as:
ω = 2πf = 2π * 60 = 120π rad/s
Now we can differentiate the magnetic flux with respect to time:
d/dt(Φ) = d/dt[(1.3 * 10^-3 T) * (π * 1.805625 * 10^-11 m^2) * sin(ωt)]
ε = -ω * (1.3 * 10^-3 T) * (π * 1.805625 * 10^-11 m^2) * cos(ωt)
Since we are looking for the maximum value, we can assume that cos(ωt) is at its maximum value of 1. Therefore, the maximum emf will be:
ε_max = ω * (1.3 * 10^-3 T) * (π * 1.805625 * 10^-11 m^2)
Now, plug in the values:
ε_max = (120π rad/s) * (1.3 * 10^-3 T) * (π * 1.805625 * 10^-11 m^2)
ε_max ≈ 2.594 * 10^-10 V
Therefore, the maximum emf that can be generated around the perimeter of the red blood cell is approximately 2.594 * 10^-10 V.
http://en.wikipedia.org/wiki/Electromagnetic_induction
So the flux is B*area
Assume B is 1.3E-3 sinwt where w is 2PI*60.
Take the derivative of B, then d flux/dt is equal to Area * the derivative of B.