Electromagnetic Induction And Alternating Currents – NEET UG PhysicsPractice Questions & PYQs

The total magnetic flux linked with a multi-turn solenoid is the product of the number of turns and the flux through each individual turn, represented as $\phi = N \Phi$. Self-inductance is defined as the ratio of total magnetic flux to the current flowing through the circuit, expressed by the formula $L = \frac{\phi}{I}$. Substituting the total flux and current values into this equation gives $L = \frac{500 \times 4 \times 10^{-3}}{2}$, which calculates to $1.0 \text{ henry}$.

The average power in an alternating current circuit is determined by integrating the instantaneous power, which is the product of voltage and current, over one complete cycle. Substituting the given sinusoidal expressions for electromotive force and current and applying trigonometric product-to-sum identities yields a time-dependent power expression containing an oscillating term and a constant term. When averaged over a full period, the oscillating component vanishes to zero, leaving only the constant term that accounts for the phase difference between the waveforms. This results in the standard expression $P_{\text{avg}} = \frac{E_0 I_0}{2} \cos \phi$, where the cosine factor represents the power factor of the circuit.

Current reaches its maximum value in a series LCR circuit under resonance conditions, where the angular frequency is determined by the resonant relation $\omega = \frac{1}{\sqrt{LC}}$. Rearranging this equation to solve for the unknown inductance yields $L = \frac{1}{\omega^2 C}$. Substituting the given angular frequency of one thousand inverse seconds and capacitance of ten microfarads into this expression gives $L = \frac{1}{(1000)^2 × 10 × 10^{-6}}$, which simplifies to zero point one henries, or one hundred millihenries.

Transformer efficiency is defined as the ratio of useful power output to total electrical power input, expressed by the formula $\eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100$. The power output of the lamp is given as one hundred watts, whereas the electrical power supplied at the primary is the product of the mains voltage and primary current, yielding two hundred twenty volts multiplied by zero point five amperes, which equals one hundred ten watts. Substituting these power values into the efficiency relation gives a ratio of one hundred over one hundred ten, resulting in an efficiency of approximately ninety percent.

Faraday's law dictates that the induced electromotive force in a coil equals the rate of change of magnetic flux with respect to time, allowing the primary voltage to be found by differentiating the given flux expression. Differentiating the flux function with respect to time yields a constant primary voltage of four volts. Applying the transformer turns ratio formula $\frac{V_s}{V_p} = \frac{N_s}{N_p}$, where the secondary turns are fifteen hundred and primary turns are fifty, relates the primary and secondary voltages. Multiplying the primary voltage by this turns ratio factor results in an output voltage of one hundred twenty volts.

The mutual inductance between two magnetically coupled coils depends on their individual self-inductances and the magnetic coupling coefficient, expressed through the relation $M = K \sqrt{L_1 L_2}$. When the effective flux from one coil completely links with the other, the coupling coefficient reaches its maximum ideal value of one. Substituting the given self-inductances and coupling coefficient into the formula, we find $M = 1 \times \sqrt{2 \times 8}$, which simplifies to the square root of sixteen, resulting in four millihenries.

Alternating magnetic fields inside a transformer core induce circulating currents within the metal body, which generate unwanted thermal energy and reduce overall electrical efficiency. By dividing the core into thin, insulated sheets, the physical pathways available for these internal currents are severely restricted. This structural technique successfully suppresses the circulating currents, thereby minimizing energy losses due to eddy currents during operation.

The characteristic frequency of an electrical oscillation in a resonant circuit is inversely proportional to the square root of the product of inductance and capacitance, represented by the formula $f = \frac{1}{2 \pi \sqrt{LC}}$. When the inductance is doubled and the capacitance is quadrupled, the new product under the square root becomes eight times the original value. Taking the inverse square root of this change scales the original frequency down by a factor of the square root of eight. Simplifying this radical yields a new frequency equal to the original frequency divided by two times the square root of two.

The power factor of an alternating current circuit is defined as the ratio of resistance to impedance, expressed mathematically as $\cos \phi = \frac{R}{Z}$. The impedance of a series circuit containing resistance, inductance, and capacitance is given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$. Substituting the given values, the impedance becomes $Z = \sqrt{8^2 + (31 - 25)^2}$, which simplifies to ten ohms. Taking the ratio of the resistance to this calculated impedance yields eight-tenths, which equals $0.80$.

The phase relationship between the alternating current and voltage in a series circuit containing resistance, inductance, and capacitance is governed by the tangent of the phase angle, expressed as $\tan \phi = \frac{X_C - X_L}{R}$. Given that the current leads the voltage by forty-five degrees, the tangent of this phase angle equals one, making the difference between capacitive and inductive reactances equal to the resistance. Substituting the angular frequency definitions for reactances and rearranging the algebraic expression to isolate capacitance yields the required formula. This derivation results in the capacitance being equal to one divided by the product of two pi times frequency and the sum of inductive reactance and resistance.

Electromotive force is fundamentally defined as the work done per unit charge in circulating a test charge once around a closed loop. Because an induced voltage $V$ acts as the effective potential driving the charge along the circuit, the total work done is calculated as the product of the charge and the induced voltage. Therefore, moving a charge $Q$ once around the loop requires a total work of $W = Q V$.

The time constant of an $LR$ series circuit is given by the formula $\tau = \frac{L}{R}$. Substituting the inductance of 40 H and resistance of 8 ohms into this expression yields 40 divided by 8, resulting in a time constant of 5 seconds.

Faraday's law of electromagnetic induction states that the magnitude of induced electromotive force is equal to the rate of change of magnetic flux, expressed as $V = \frac{\Delta \phi}{\Delta t}$. Since electric current is the rate of flow of charge, we can write the current as $I = \frac{Q}{\Delta t}$, which by Ohm's law equals the induced voltage divided by resistance $R$. Equating these expressions and canceling the time interval from both sides yields the total charge as the change in flux divided by the resistance.

In an alternating current series $LCR$ circuit, resonance occurs when inductive reactance equals capacitive reactance, making the total impedance of the circuit purely resistive and equal to the resistance $R$. Because the phase angle between voltage and current becomes zero at resonance, the power factor is unity and the active power loss in the circuit is simply determined by the square of the root-mean-square current multiplied by this pure resistance, yielding $I^2 R$.

The induced electromotive force in an inductor is directly proportional to the time rate of change of current, meaning the emf becomes zero precisely when the derivative of current with respect to time is zero. Applying the product rule to differentiate the given current expression $I = t^2 e^{-t}$ yields an expression containing $2t e^{-t} - t^2 e^{-t}$, and setting this derivative equal to zero factors out to $t - 2 = 0$, which directly points to a time of 2 seconds.

Capacitive reactance is inversely proportional to both the angular frequency of the supply and the capacitance of the capacitor, following the relation $X = \frac{1}{\omega C}$. When both the frequency and the capacitance are doubled, the new reactance is calculated by substituting $2\omega$ for frequency and $2C$ for capacitance into the denominator, introducing a factor of four that reduces the initial reactance to one-fourth of its original value.

The root-mean-square current in an alternating current circuit containing a capacitor depends on the RMS voltage and the capacitive reactance, defined by the formula $X_C = \frac{1}{\omega C}$. Comparing the voltage equation to the standard form gives a peak voltage of $200\sqrt{2}\,\text{V}$ and an angular frequency of $100\,\text{rad/s}$, leading to an RMS voltage of $200\,\text{V}$ and a capacitive reactance of $10,000\,\Omega$ for a $1\,\mu\text{F}$ capacitor, which results in an RMS current of $20\,\text{mA}$ upon division.

Capacitors behave as a short circuit at the exact instant a circuit is energized and transition into an open circuit once fully charged in the steady state. At the moment the key is pressed, the uncharged capacitor permits current to pass freely through its branch, bypassing the parallel resistance path. Consequently, the active circuit resistance is restricted to the single resistor $AB$, establishing an initial current of 2 mA.

As time progresses, the capacitor accumulates charge and ceases current flow entirely, forcing the circuit into its steady-state configuration. The current must now travel through the total series resistance of 2000 ohms, which reduces the current through the resistor to 1 mA according to $I = \frac{V}{R}$.

Average power dissipation in an alternating current circuit is calculated using the root-mean-square values of voltage and current alongside the phase angle through the relation $P = I_{\text{rms}} E_{\text{rms}} \cos \phi$. Extracting the peak values from the given equations yields $i_0 = \frac{1}{\sqrt{2}}$ A and $e_0 = \frac{1}{\sqrt{2}}$ V, which convert to root-mean-square values of $I_{\text{rms}} = \frac{1}{2}$ A and $E_{\text{rms}} = \frac{1}{2}$ V. Substituting these quantities and the phase difference of $\phi = \frac{\pi}{3}$ into the power formula gives $P = \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) \cos\left(\frac{\pi}{3}\right)$, evaluating to $\frac{1}{8}$ watts.

The quality factor of a resonant circuit is defined by the ratio of reactive impedance to resistance, which can be expressed in terms of capacitance, inductance, and angular frequency. At resonance, the inductive reactance equals the capacitive reactance, allowing the quality factor formula $\frac{\omega L}{R}$ to be equivalently rewritten by substituting $\omega^2 = \frac{1}{L C}$. This algebraic substitution transforms the expression into $\frac{1}{\omega R C}$, which directly corresponds to the correct alternative.