Need help on which formulas I use to solve problems?
A coil of wire with 200 circular turns of radius 3.00 cm is in a uniform magnetic field along the axis of the coil. The coil has R = 40.0 ohms. At what rate, in teslas per second, must the magnetic field be changing to induce a current of 0.150 A in the coil?
What is the reactant of a 3.00 H inductor at a frequency of 80.0 Hz?
Compute the impedance of a series R-L-C circuit at angular frequencies of 1000, 750, and 500 rad/s. Take R = 200, L = 0.900 H, and C = 2.00 F.
10.6 t/s
I need to have the answer for the same question
To solve these problems, you can use the following formulas:
1) Faraday's Law of Electromagnetic Induction:
ε = -dΦ/dt
ε is the induced emf (in volts)
Φ is the magnetic flux (in webers)
t is time (in seconds)
Rearranging the formula, we have:
dΦ/dt = -ε
2) Ohm's Law:
V = I * R
V is the voltage (in volts)
I is the current (in amperes)
R is the resistance (in ohms)
3) Impedance of an R-L-C circuit:
Z = √(R^2 + (ωL - 1/(ωC))^2)
Z is the impedance (in ohms)
R is the resistance (in ohms)
L is the inductance (in henries)
C is the capacitance (in farads)
ω is the angular frequency (in radians per second)
Now, let's solve each problem step by step:
Problem 1:
Given: N = 200, r = 3.00 cm = 0.03 m, R = 40.0 ohms, I = 0.150 A
We need to find the rate of change of the magnetic field (dΦ/dt).
1) Calculate the area of one loop of the coil:
A = π * r^2
2) Calculate the change in magnetic flux:
dΦ = N * B * A
3) Substitute the given values to find the change in magnetic flux:
dΦ = 200 * B * (π * (0.03)^2)
4) Rearrange Faraday's Law of Electromagnetic Induction to solve for the rate of change of the magnetic field:
-dΦ/dt = ε
dΦ/dt = -ε
5) Substitute the given values to find the rate of change of the magnetic field:
dΦ/dt = -0.150
Therefore, the rate of change of the magnetic field must be -0.150 T/s.
Problem 2:
Given: L = 3.00 H, f = 80.0 Hz
We need to find the reactance of the inductor at a frequency of 80.0 Hz.
1) Calculate the angular frequency:
ω = 2πf
2) Substitute the given values to find the reactance of the inductor:
X_L = ωL
3) Substitute the given values to find the reactance:
X_L = 2π * 80.0 * 3.00
Therefore, the reactance of the 3.00 H inductor at a frequency of 80.0 Hz is 1506.72 ohms.
Problem 3:
Given: R = 200 ohms, L = 0.900 H, C = 2.00 F, ω = 1000 rad/s, 750 rad/s, 500 rad/s
We need to find the impedance of the series R-L-C circuit at different angular frequencies.
1) Substitute the given values into the impedance formula to calculate the impedance at ω = 1000 rad/s:
Z = √(R^2 + (ωL - 1/(ωC))^2)
Z = √(200^2 + (1000 * 0.900 - 1/(1000 * 2.00))^2)
2) Calculate the impedance at ω = 750 rad/s and ω = 500 rad/s using the same formula.
Therefore, the impedance of the series R-L-C circuit at angular frequencies of 1000 rad/s, 750 rad/s, and 500 rad/s can be calculated by substituting the given values into the impedance formula.
To solve these problems, you will need to use different formulas from the field of electromagnetism and circuits. Here are the formulas you can use for each problem:
1. To find the rate at which the magnetic field must be changing to induce a current in the coil:
- Faraday's Law of Electromagnetic Induction: ε = -N * [(ΔΦ) / Δt]
- This formula relates the induced electromotive force (ε) to the change in magnetic flux (ΔΦ) over time (Δt).
- N = Number of turns in the coil
- ΔΦ = Change in magnetic flux
- Δt = Change in time
In this problem, we are given N = 200 (number of turns) and a current of 0.150 A. We need to find ΔΦ/Δt, which represents the rate of change of magnetic flux required. Rearranging the formula, we get:
(ΔΦ/Δt) = -(ε / N) = -I / N
Here, I is the current flowing through the coil.
2. To find the reactance of an inductor at a given frequency:
- Inductive Reactance: XL = 2πfL
- This formula calculates the reactance (XL) of an inductor at a given frequency (f) and inductance (L).
- XL = Inductive reactance
- f = Frequency
- L = Inductance
In this problem, we are given the inductance L = 3.00 H and frequency f = 80.0 Hz. Plugging these values into the formula, we can calculate the reactance (XL) of the inductor.
3. To compute the impedance of a series R-L-C circuit at different angular frequencies:
- Impedance: Z = √(R^2 + (XL - XC)^2)
- This formula calculates the total impedance (Z) of a series R-L-C circuit, where R is the resistance, XL is the inductive reactance, and XC is the capacitive reactance.
- Z = Impedance
- R = Resistance
- XL = Inductive reactance
- XC = Capacitive reactance
In this problem, we are given the values R = 200, L = 0.900 H, and C = 2.00 F. We need to calculate the impedance (Z) at different angular frequencies (ω). The inductive reactance (XL) and capacitive reactance (XC) can be calculated using their respective formulas:
- XL = 2πfL
- XC = 1 / (2πfC)
Plugging these values into the impedance formula, we can calculate the impedance at different angular frequencies.