A series LCR circuit connected to a variable frequency 230 V source. L = 5.0 H, C = 80mF, R = 40 Ω.
(a) Determine the source frequency which drives the circuit in resonance.
(b) Obtain the impedance of the circuit and the amplitude of current at the resonating frequency.
(c) Determine the rms potential drops across the three elements of the circuit. Show that the potential drop across the LC combination is zero at the resonating frequency
Inductance of the inductor, L = 5.0 H Capacitance of the capacitor, C = 80 µH = 80 × 10−6 F Resistance of the resistor, R = 40 Ω Potential of the variable voltage source, V = 230 V (a) Resonance angular frequency is given as:
\(ωr=\frac{1}{√LC}=\frac{1}{√5X80X10^{-6}}=\frac{10^3}{20}=50 rad/s\)
Hence, the circuit will come in resonance for a source frequency of 50 rad/s. (b) Impedance of the circuit is given by the relation:
\(Z=\sqrt(R^2+(X_L-X_C)^2)\)At resonance, \(X_L=X_C,Z=R=40 Ω\)
Amplitude of the current at the resonating frequency is given as:\(I°=\frac{V°}{Z}\)
Where, V°=Peak voltage = \(√2V=I°=\frac{√2V}{Z}=\frac{√2×230}{40}=8.13 A\)
Hence, at resonance, the impedance of the circuit is 40 Ω and the amplitude of the current is 8.13 A. (c) rms potential drop across the inductor,
(VL)rms=IxωrL Where,
\(Irms=\frac{I°}{√2}=\frac{√2V}{√2Z}=\frac{230}{40}=\frac{23}{4}A=(VL)rms=\frac{23}{4}x50x5=1437.5V\)
Potential drop across the capacitor:
\((V_c)rms=I=\frac{1}{w_rc}=\frac{23}{4}X\frac{23}{4}X\frac{1}{50X80X10^{-6}}=1437.5V=230V\)
Potential drop across the resistor:
\((VR)rms=IR=\frac{23}{4}X40=230V\)
Potential drop across the LC combination:
\(VLC=I(X_L-X_C)\)
At resonance, \(X_L=X_C,V_LC=0\)
Hence, it is proved that the potential drop across the LC combination is zero at resonating frequency.
Three students, Neha, Rani, and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads, and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads, and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads, and 3 erasers.
Based upon the above information, answer the following questions:
(i) Form the equations required to solve the problem of finding the price of each item, and express it in the matrix form \( A \mathbf{X} = B \).
Simar, Tanvi, and Umara were partners in a firm sharing profits and losses in the ratio of 5 : 6 : 9. On 31st March, 2024, their Balance Sheet was as follows:
Liabilities | Amount (₹) | Assets | Amount (₹) |
Capitals: | Fixed Assets | 25,00,000 | |
Simar | 13,00,000 | Stock | 10,00,000 |
Tanvi | 12,00,000 | Debtors | 8,00,000 |
Umara | 14,00,000 | Cash | 7,00,000 |
General Reserve | 7,00,000 | Profit and Loss A/c | 2,00,000 |
Trade Payables | 6,00,000 | ||
Total | 52,00,000 | Total | 52,00,000 |
Umara died on 30th June, 2024. The partnership deed provided for the following on the death of a partner:
A coil of 60 turns and area \( 1.5 \times 10^{-3} \, \text{m}^2 \) carrying a current of 2 A lies in a vertical plane. It experiences a torque of 0.12 Nm when placed in a uniform horizontal magnetic field. The torque acting on the coil changes to 0.05 Nm after the coil is rotated about its diameter by 90°. Find the magnitude of the magnetic field.
The sequence of nitrogenous bases in a segment of a coding strand of DNA is
5' – AATGCTAGGCAC – 3'. Choose the option that shows the correct sequence of nitrogenous bases in the mRNA transcribed by the DNA.
An LCR circuit, also known as a resonant circuit, or an RLC circuit, is an electrical circuit consist of an inductor (L), capacitor (C) and resistor (R) connected in series or parallel.
When a constant voltage source is connected across a resistor a current is induced in it. This current has a unique direction and flows from the negative to positive terminal. Magnitude of current remains constant.
Alternating current is the current if the direction of current through this resistor changes periodically. An AC generator or AC dynamo can be used as AC voltage source.