A 34.7 cm diameter coil consists of 20 turns of circular copper wire 1.50 mm in diameter. A uniform magnetic field, perpendicular to the plane of the coil, changes at a rate of 9.00E-3 T/s. Determine the current in the loop.
Determine the rate at which thermal energy is produced.
To determine the current in the loop, we can use Faraday's Law of electromagnetic induction, which states that the induced electromotive force (EMF) in a closed loop is equal to the rate of change of magnetic flux through the loop.
First, let's calculate the magnetic flux through the loop. The magnetic flux (Φ) is given by the formula:
Φ = B * A
Where:
- B is the magnetic field strength (9.00E-3 T/s)
- A is the area of the loop
The area of the loop can be calculated as the product of the number of turns (N) and the area of each turn (a):
A = N * a
Where:
- N is the number of turns (20)
- a is the area of each turn
The area of each turn can be calculated as the circular area:
a = π * r^2
Where:
- r is the radius of each turn (which is half the diameter)
Given the diameter of the coil is 34.7 cm, the radius (r) can be calculated as:
r = 34.7 cm / 2
Now we have all the values needed to calculate the area of each turn (a) and the magnetic flux (Φ).
Once we have Φ, we can calculate the induced EMF (ε) using the formula:
ε = -dΦ/dt
Where:
- dΦ/dt is the rate of change of magnetic flux (9.00E-3 T/s)
Finally, we can use Ohm's Law (V = I * R) to find the current (I) in the loop, where V is the induced EMF and R is the resistance of the loop. Since the coil is made of copper, we can use the resistivity of copper to find the resistance.
Now, let's go step by step to calculate the current in the loop:
1. Calculate the radius of each turn:
- r = 34.7 cm / 2 = 17.35 cm = 0.1735 m
2. Calculate the area of each turn:
- a = π * r^2 = 3.14159 * (0.1735 m)^2 = 0.094201 m^2
3. Calculate the area of the loop:
- A = N * a = 20 * 0.094201 m^2 = 1.88402 m^2
4. Calculate the magnetic flux through the loop:
- Φ = B * A = (9.00E-3 T/s) * 1.88402 m^2 = 0.01695618 T·m^2
5. Calculate the induced EMF:
- ε = -dΦ/dt = -(0.01695618 T·m^2) / (1 s) = -0.01695618 V
6. Calculate the current in the loop:
- V = I * R
- I = V / R
To find the resistance of the loop, we need the length and the cross-sectional area of the wire. The length of the wire can be calculated using the number of turns, the length of each turn, and the thickness of the wire.
Let's assume the length of each turn is equal to the circumference of the coil, which can be calculated as:
C = 2π * r
The total length of the wire is then given by:
L = N * C
Where:
- N is the number of turns (20)
- C is the circumference of each turn
The circumference of each turn can be calculated using the wire diameter (1.50 mm) as:
C = π * d
Where:
- d is the diameter of the wire
Now we can calculate the length of the wire (L). Finally, we can find the cross-sectional area (cross-sectional area of a wire) and the resistance (R) using the formula:
R = resistivity * (L / A)
Where:
- resistivity is the resistivity of copper (1.72E-8 Ω·m)
To calculate the rate at which thermal energy is produced, we can use the formula:
P = I^2 * R
Where:
- P is the power (rate at which thermal energy is produced)
- I is the current in the loop
- R is the resistance of the loop
Now, let's calculate the current in the loop and the rate at which thermal energy is produced.