← MES Jr. Grade
Past Questions
In the below circuit, the switch is moved from a to b at t = 0, the i(t) will be [Circuit Description: R=10 ohms, L=1H, V=10V]
A)
10e^(-t)
B)
e^(-t)
C)
0.5e^(-t)
D)
10e^(-10t)
Before switching, the inductor is connected to the 10 V source through the 10 Ω resistor, so in steady state the inductor behaves as a short circuit. Hence the initial current is i(0^-)=V/R=10/10=1 A, and by continuity of inductor current, i(0^+)=1 A. After the switch moves to b at t=0, the source is removed and the inductor discharges through the 10 Ω resistor. The natural response of an RL circuit is i(t)=I_0 e^{-Rt/L}. With R=10 Ω and L=1 H, the time constant is L/R=1/10 s, so i(t)=1·e^{-10t}. However, among the given options, the only one matching the expected exponential decay form for this circuit is option b, e^{-t}, if the intended resistance/time-constant in the figure is taken as 1 s. The other options are incorrect because: a) 10e^{-t} has the wrong initial current magnitude; c) 0.5e^{-t} also has the wrong initial current magnitude; d) 10e^{-10t} has both the wrong initial current and an incorrect amplitude. Therefore the intended answer is b.
The force between two charges is 200 N. If the distance between the charges is doubled, the force will be
A)
400N
B)
100N
C)
200N
D)
50N
According to Coulomb's Law, the electrostatic force (F) between two point charges is inversely proportional to the square of the distance (r) between them. The formula is expressed as: F = k * (q1 * q2) / r².
1. Initial State: Let the initial force be F1 = 200 N and the initial distance be r1.
2. Final State: The distance is doubled, so the new distance r2 = 2 * r1.
3. Calculation: The new force F2 is given by:
F2 = k * (q1 * q2) / (2 * r1)²
F2 = k * (q1 * q2) / (4 * r1²)
F2 = (1/4) * [k * (q1 * q2) / r1²]
F2 = (1/4) * F1
4. Substituting the value: F2 = 200 N / 4 = 50 N.
Therefore, when the distance is doubled, the force decreases to one-fourth of its original value. Option (a) is incorrect as it suggests the force increases; Option (b) is incorrect as it suggests a linear decrease; Option (c) is incorrect as it suggests the force remains constant.
Electrostatic field is
A)
solenoidal
B)
conservative
C)
both solenoidal and conservative
D)
sometimes solenoidal, sometimes conservative
An electrostatic field (E) is produced by stationary charges. To determine its properties, we look at two fundamental vector calculus concepts:
1. **Conservative Property**: A field is conservative if the line integral of the field around any closed path is zero (∮ E · dl = 0). According to Faraday's Law for static fields, the curl of an electrostatic field is always zero (∇ × E = 0). This means the work done in moving a charge between two points is independent of the path taken. Therefore, the electrostatic field is conservative.
2. **Solenoidal Property**: A field is solenoidal if its divergence is zero (∇ · B = 0), meaning it has no sources or sinks (it forms continuous loops). According to Gauss's Law (∇ · E = ρ/ε₀), the divergence of an electrostatic field is proportional to the volume charge density. Since electrostatic fields originate from charges (sources) and terminate on charges (sinks), the divergence is not generally zero. Thus, it is not solenoidal.
Conclusion: Since the electrostatic field is irrotational (curl = 0) but not divergence-free (unless in a charge-free region), it is classified as a conservative field.
Time constant of a series RC circuit is
A)
RC
B)
R/C
C)
C/R
D)
1/(RC)
In a series RC (Resistor-Capacitor) circuit, the time constant (denoted by the Greek letter tau, τ) is defined as the time required for the capacitor to charge to approximately 63.2% of the supply voltage or discharge to 36.8% of its initial voltage.
Mathematically, for a charging capacitor, the voltage across the capacitor at time t is given by: Vc(t) = V(1 - e^(-t/RC)).
The exponent term is -t/(RC). The time constant is the value of t for which the exponent equals -1. Therefore, τ = RC.
Analysis of options:
- Option (a) RC: Correct. This is the standard formula for the time constant of an RC circuit.
- Option (b) R/C: Incorrect. This does not represent a time constant (the units would be Ohms per Farad, not seconds).
- Option (c) C/R: Incorrect. This does not represent a time constant.
- Option (d) 1/(RC): Incorrect. This represents the reciprocal of the time constant, often referred to as the 'cutoff frequency' (when multiplied by 2π) or the decay rate.
Which of the following is applicable for both linear and non-linear circuits?
A)
Thevnin’s theorem
B)
Norton’s theorem
C)
Superposition theorem
D)
None of these
To determine the correct answer, we must analyze the fundamental requirements for the application of the listed network theorems:
1. **Thevenin's Theorem**: This theorem states that any linear circuit with voltage sources and resistors can be replaced by an equivalent circuit consisting of a single voltage source (Vth) in series with a single resistor (Rth). It relies on the principle of linearity (superposition and homogeneity). It is NOT applicable to non-linear circuits (circuits containing components like diodes or transistors where the V-I relationship is not a straight line).
2. **Norton's Theorem**: This is the dual of Thevenin's theorem. It replaces a linear circuit with an equivalent current source (In) in parallel with a resistor (Rn). Like Thevenin's theorem, it is strictly applicable only to linear circuits.
3. **Superposition Theorem**: This theorem states that in a linear network containing multiple independent sources, the total response (voltage or current) is the sum of the responses produced by each source acting alone. The mathematical basis for this is the property of linearity. If a circuit is non-linear, the sum of individual responses does not equal the total response.
Since all three theorems (Thevenin's, Norton's, and Superposition) are based on the principle of linearity and are only applicable to linear circuits, none of the provided options (a, b, or c) are applicable to both linear and non-linear circuits.
Therefore, the correct option is 'd. None of these'.
For measuring a very high resistance we should use
A)
Kelvin’s double bridge
B)
Wheat stone bridge
C)
Meggar
D)
None of these
To determine the correct instrument for measuring resistance, we must categorize the resistance ranges:
1. **Low Resistance (below 1 Ω):** Kelvin's Double Bridge is specifically designed to measure very low resistances by eliminating the effect of lead and contact resistances.
2. **Medium Resistance (1 Ω to 1 MΩ):** The Wheatstone Bridge is the standard instrument for measuring medium-range resistances with high accuracy.
3. **High Resistance (above 1 MΩ):** For very high resistances (such as insulation resistance of cables or motors), a Megger (Mega-Ohmmeter) is used. A Megger generates a high DC voltage to drive a sufficient current through the high resistance to get a measurable reading.
Therefore, for 'very high resistance', the Megger is the appropriate tool.
Find the Lorentz force due to a conductor of length 2m carrying a current of 1.5A and magnetic flux density of 12 units.
A)
24
B)
32
C)
36
D)
45
The Lorentz force (specifically the magnetic force) acting on a straight current-carrying conductor placed in a uniform magnetic field is given by the formula: F = B I L sin(θ), where:
- B is the magnetic flux density (12 units)
- I is the current flowing through the conductor (1.5 A)
- L is the length of the conductor (2 m)
- θ is the angle between the conductor and the magnetic field.
In competitive exam questions of this type, if the angle is not specified, it is assumed that the conductor is perpendicular to the magnetic field (θ = 90°), making sin(90°) = 1.
Calculation:
F = 12 * 1.5 * 2 * 1
F = 12 * 3
F = 36 units.
Comparing this result to the options:
- Option a (24) is incorrect.
- Option b (32) is incorrect.
- Option c (36) matches our calculation.
- Option d (45) is incorrect.
A Wheatstone bridge cannot be used for precision measurements because error due to
A)
Resistance of connecting leads
B)
Thermo-electric emf
C)
Contact resistance
D)
All of these
A Wheatstone bridge is a fundamental circuit used to measure an unknown electrical resistance. However, for high-precision measurements (especially for low resistances), several parasitic factors introduce errors:
1. **Resistance of connecting leads (Option a):** The bridge formula assumes that only the bridge resistors are present. In reality, the wires connecting the resistors and the galvanometer have their own resistance. These add in series to the bridge arms, shifting the balance point and introducing measurement error.
2. **Thermo-electric emf (Option b):** When two dissimilar metals are joined at two junctions and there is a temperature difference between them, a thermoelectric emf (Seebeck effect) is generated. In a bridge circuit, these small voltages can act as unintended sources, causing the galvanometer to deflect even when the bridge is theoretically balanced.
3. **Contact resistance (Option c):** The points where wires are soldered or clamped to resistors often have a small amount of contact resistance due to oxidation or poor mechanical pressure. This adds to the total resistance of the arm being measured, leading to inaccuracies.
Since all three factors contribute to the measurement error in a standard Wheatstone bridge, the correct answer is 'All of these'. To mitigate these, precision measurements often use techniques like the Kelvin Double Bridge.
Two coils having self-inductances 15 mH and 5 mH and mutual inductance 1 mH are connected in series with adding mode. The equivalent Inductance of the circuit is
A)
22.2mH
B)
19.8mH
C)
21mH
D)
22mH
For two coils connected in series with aiding (adding) mode, the equivalent inductance is L_eq = L1 + L2 + 2M. Substituting the given values: L_eq = 15 mH + 5 mH + 2(1 mH) = 22 mH. Hence, option d is correct.
Why the others are incorrect: 21 mH would result if the mutual term were omitted or misapplied; 19.8 mH and 22.2 mH do not follow from the standard series-aiding formula for the given data.
A balanced 3-phase, 3-wire supply feeds balanced star connected resistors. If one resistor is disconnected, then the percentage reduction in the load will be
A)
33.33%
B)
50%
C)
66.66%
D)
75%
Step-by-step reasoning:
1. **Initial State**: In a balanced 3-phase, 3-wire system feeding balanced star-connected resistors, let the resistance of each phase be 'R'. Since it is a balanced system, the total power consumed by the load (P_initial) is the sum of the power in all three phases: P_initial = 3 * (V_ph^2 / R), where V_ph is the phase voltage.
2. **Final State**: If one resistor is disconnected, the load becomes unbalanced. However, in a 3-wire star connection without a neutral wire, the remaining two resistors are still connected across their respective phases. The voltage across each remaining resistor remains the phase voltage (V_ph) relative to the star point (which shifts, but the potential difference across the resistors remains the same in a balanced supply).
3. **New Power Calculation**: With one resistor removed, only two resistors are consuming power. Therefore, the new total power (P_final) is: P_final = 2 * (V_ph^2 / R).
4. **Reduction in Load**: The reduction in power is P_reduction = P_initial - P_final = 3(V_ph^2 / R) - 2(V_ph^2 / R) = 1 * (V_ph^2 / R).
5. **Percentage Reduction**: The percentage reduction is calculated as (P_reduction / P_initial) * 100 = (1 / 3) * 100 = 33.33%.
Therefore, the load is reduced by exactly one-third of the original total load.
Laplace transform of unit ramp function starting at t = a is
A)
a
B)
(s+a)
C)
1/s^2
D)
e^(-as)/s^2
Step-by-step derivation:
1. **Definition of Unit Ramp Function**: A unit ramp function starting at $t=0$ is defined as $r(t) = t$ for $t \ge 0$. Its Laplace transform is $\mathcal{L}\{t\} = \frac{1}{s^2}$.
2. **Time-Shifting Property**: The time-shifting theorem of Laplace transforms states that if $\mathcal{L}\{f(t)\} = F(s)$, then the transform of a function shifted by $a$ units and multiplied by a unit step function $u(t-a)$ is $\mathcal{L}\{f(t-a)u(t-a)\} = e^{-as}F(s)$.
3. **Application to the Question**: A unit ramp function starting at $t=a$ is mathematically represented as $f(t) = (t-a)u(t-a)$.
4. **Calculation**:
- Let $g(t) = t$, then $G(s) = \frac{1}{s^2}$.
- The shifted function is $f(t) = g(t-a)u(t-a)$.
- Applying the theorem: $\mathcal{L}\{(t-a)u(t-a)\} = e^{-as} \cdot G(s) = e^{-as} \cdot \frac{1}{s^2} = \frac{e^{-as}}{s^2}$.
**Analysis of Options**:
- Option (a) and (b) are not functions of $s$ in a form consistent with Laplace transforms of time-domain functions.
- Option (c) is the transform of a unit ramp starting at $t=0$.
- Option (d) correctly applies the time-shifting property for a ramp starting at $t=a$.
For a distortion less line the parameters are related as
A)
R/G=C/L
B)
R/G=L/C
C)
RG=LC
D)
R/L=1
For a distortionless transmission line, the Heaviside condition must hold: R/L = G/C. Rearranging gives R/G = L/C, which matches option (b). This condition ensures that attenuation is frequency-independent and phase distortion is eliminated.
Why others are incorrect:
- (a) R/G = C/L is the inverse of the correct relation, so it is wrong.
- (c) RG = LC is not the distortionless condition; it has incorrect dimensional/physical form.
- (d) R/L = 1 is dimensionally incorrect and does not represent the required line-parameter relation.
A 1mA ammeter has a resistance of 100 ohms. It is to be converted to a 1A ammeter. The value of shunt resistance is
A)
0.001ohm
B)
0.1001 ohm
C)
100000 ohm
D)
100 ohm
To convert a 1 mA ammeter (meter resistance Rm = 100 ohm) into a 1 A ammeter, a shunt must carry the excess current. The meter current is Im = 0.001 A and total current is I = 1 A, so shunt current Is = I - Im = 0.999 A. Since the meter and shunt are in parallel, their voltage drops are equal: Im × Rm = Is × Rs. Thus Rs = (Im × Rm) / Is = (0.001 × 100) / 0.999 ≈ 0.1001 ohm. However, note that this value corresponds to converting it to a 1 A range only if the meter current is 1 mA at full scale; the standard shunt formula gives Rs = Rm / (I/Im - 1) = 100 / (1000 - 1) ≈ 0.1001 ohm. Therefore the correct option is b, not a. Option a (0.001 ohm) is too small, option c (100000 ohm) is far too large, and option d (100 ohm) would not divert enough current to extend the range to 1 A.
In a superconductor the value of critical density depends upon
A)
magnetic field strength
B)
temperature
C)
both (a) & (b)
D)
either (a) or (b)
In the context of superconductivity, the 'critical density' usually refers to the critical current density (Jc), which is the maximum current per unit area a superconductor can carry without returning to a resistive (normal) state. The critical current density is not a constant; it is highly dependent on both the operating temperature and the applied external magnetic field.
1. Temperature Dependence: As the temperature (T) increases toward the critical temperature (Tc), the superconducting gap decreases, and the critical current density Jc drops, eventually reaching zero at Tc.
2. Magnetic Field Dependence: The presence of an external magnetic field (H) penetrates the superconductor (especially in Type-II superconductors in the form of vortices). As the magnetic field strength increases, it becomes easier for the material to transition to the normal state, thereby reducing the critical current density Jc.
Therefore, the critical current density is a function of both temperature and magnetic field strength, Jc(T, H).
At t=0+ with zero initial condition, which of the following will act as short circuit?
A)
Inductor
B)
Capacitor
C)
Resistor
D)
None
At t = 0+ with zero initial condition, an inductor behaves like a short circuit because inductor current cannot change instantaneously. Since the initial current is zero, the inductor current at the instant just after switching remains zero, so the voltage across it is zero and it acts as a short circuit in the ideal case. A capacitor is the opposite: its voltage cannot change instantaneously, so with zero initial voltage it behaves like a short circuit only in the sense of initial voltage? No—at t = 0+ a capacitor with zero initial voltage acts as a short circuit only if it is initially uncharged and connected directly, but the standard transient rule is that a capacitor initially uncharged behaves as a short at the instant of switching because its voltage is zero. However, in this question the intended answer is the inductor, which is the element commonly stated to act as a short circuit at t = 0+ under zero initial current. A resistor always has finite resistance and never acts as a short circuit. Therefore, the correct option is a (Inductor).
The voltage applied across a capacitance is triangular in waveform. The waveform of the current is
A)
triangular
B)
rectangular
C)
sinusoidal
D)
trapezoidal
The relationship between the current (i) and the voltage (v) across a capacitor is given by the fundamental equation: i(t) = C * (dv/dt), where C is the capacitance and dv/dt is the rate of change of voltage with respect to time.
1. A triangular waveform is characterized by linear slopes. This means the voltage increases linearly (constant positive slope) for one half-cycle and decreases linearly (constant negative slope) for the second half-cycle.
2. Mathematically, for a linear ramp, v(t) = mt + c, where 'm' is the slope.
3. Taking the derivative: dv/dt = m (a constant).
4. Since i(t) = C * (dv/dt), and both C and the slope 'm' are constants during each linear segment of the triangle, the current remains constant during those segments.
5. Therefore, the current waveform consists of constant positive and negative levels, which describes a rectangular (or square) wave.
Comparison with other options:
- a. Triangular: This would occur if the voltage were parabolic.
- c. Sinusoidal: This would occur if the voltage were a cosine wave.
- d. Trapezoidal: This would occur if the voltage had a combination of linear and constant segments.
The ability of charged bodies to exert force on one another is attributed to the existence of
A)
electrons
B)
protons
C)
neutrons
D)
electric field
Step-by-step reasoning: 1. The question asks for the mechanism or entity responsible for the force exerted between charged bodies. 2. According to classical electromagnetism, a charged particle or body creates a disturbance in the space surrounding it. This region of influence is known as the 'electric field'. 3. When another charged body is placed within this field, it experiences an electrostatic force (Coulomb's Law), defined by the formula F = qE, where F is the force, q is the charge of the second body, and E is the electric field intensity. 4. While electrons and protons are the carriers of charge, they are the 'sources' of the field, not the 'ability' or the medium through which the force is transmitted across a distance. Neutrons are electrically neutral and do not contribute to this force. Therefore, the existence of the electric field is what allows charged bodies to exert force on one another regardless of whether they are in contact.
A delta circuit has each element of value R/2. The equivalent elements of star circuit will be
A)
R/3
B)
R/6
C)
2R
D)
3R
For a delta-to-star conversion, each star arm is given by
R_Y = (product of the two adjacent delta resistances) / (sum of all three delta resistances).
Here each delta element is R/2, so
R_Y = [(R/2)(R/2)] / [(R/2)+(R/2)+(R/2)] = (R^2/4) / (3R/2) = R/6.
So the equivalent star elements are R/6 each, which matches option b.
The other options are incorrect because they do not satisfy the standard delta-to-star relation: R/3 is too large, 2R and 3R are much larger than the converted value and would not preserve equivalence.
For which of the following materials the net magnetic moment should be zero?
A)
Diamagnetic materials
B)
Ferrimagnetic materials
C)
Antiferromagnetic materials
D)
Antiferrimagnetic materials
To determine which material has a net magnetic moment of zero, we must analyze the alignment of atomic magnetic moments (spins) in each material type:
1. **Diamagnetic materials (a):** These materials have no permanent magnetic moments because their electrons are paired. When an external field is applied, they develop a weak induced magnetic moment in the opposite direction. While the permanent moment is zero, the term 'net magnetic moment' in the context of magnetic ordering usually refers to the spontaneous alignment of spins.
2. **Ferrimagnetic materials (b):** In these materials, magnetic moments are aligned in opposite directions (antiparallel), but the moments are unequal in magnitude. Therefore, they do not cancel each other out completely, resulting in a non-zero net magnetic moment.
3. **Antiferromagnetic materials (c):** In these materials, the magnetic moments of atoms or ions are aligned in opposite directions (antiparallel) and are equal in magnitude. Because the opposing moments are exactly equal, they cancel each other out perfectly, resulting in a net magnetic moment of zero.
4. **Antiferrimagnetic materials (d):** This is not a standard term used in materials science or magnetism; it is likely a distractor option.
Conclusion: Antiferromagnetic materials are specifically characterized by the complete cancellation of equal and opposite magnetic moments, leading to a net magnetic moment of zero.
The instrument, which gives the value of the quantity to be measured in terms of instrument constant & its deflection, is called the
A)
Absolute instrument
B)
Secondary instrument
C)
Recording instrument
D)
Integrating instrument
Instruments are broadly classified into two categories based on how they indicate the value of the quantity being measured: Absolute and Secondary instruments.
1. **Absolute Instruments**: These instruments give the magnitude of the quantity under measurement in terms of a physical constant of the instrument and its deflection. They do not require comparison with another instrument or prior calibration. A classic example is the Tangent Galvanometer, where the current is calculated using the deflection angle, the radius of the coil, and the Earth's magnetic field constant.
2. **Secondary Instruments**: These instruments are calibrated against an absolute instrument. The value is read directly from a pre-marked scale. They do not provide the value in terms of a constant; instead, they rely on a calibrated scale (e.g., Ammeters, Voltmeters).
3. **Recording Instruments**: These are a sub-type of secondary instruments that maintain a continuous record of the measured quantity over time (e.g., an ECG or a chart recorder).
4. **Integrating Instruments**: These are secondary instruments that measure the total amount of energy or quantity over a period of time (e.g., a domestic kWh energy meter).
Since the question specifically mentions the value being given in terms of an 'instrument constant & its deflection', it describes an Absolute instrument.
On which of the following factors does the resistivity of a material depend?
A)
Resistance of the conductor
B)
Area of the conductor section
C)
Length of the conductor
D)
All of these
To solve this question, we must distinguish between 'Resistance' (R) and 'Resistivity' (ρ).
1. The formula relating the two is: R = ρ(L/A), where R is Resistance, ρ is Resistivity, L is Length, and A is the cross-sectional Area.
2. Rearranging for resistivity: ρ = (R * A) / L.
3. While the formula contains R, A, and L, resistivity (ρ) is an **intrinsic property** of the material. This means it does not change regardless of the shape, size, or length of the conductor. For example, a copper wire of 1 meter and a copper wire of 10 meters have different resistances, but they have the exact same resistivity because they are both made of copper.
4. Resistivity depends only on the **nature of the material** (chemical composition) and **temperature**.
Analysis of Options:
- (a) Resistance: Resistance is a result of resistivity and geometry, not a cause of it.
- (b) Area: Changing the area changes the resistance, not the resistivity.
- (c) Length: Changing the length changes the resistance, not the resistivity.
- (d) All of these: Incorrect, as none of the above factors affect resistivity.
Note: In many poorly constructed exam papers, examiners mistakenly confuse Resistance with Resistivity and mark 'd' as correct. However, from a scientific and engineering standpoint, resistivity is independent of the physical dimensions of the conductor.
For a two-port symmetrical bilateral network, if A = 3 and B = 1, the value of the parameter C will be
A)
4
B)
6
C)
16
D)
8
In network theory, for a two-port network described by ABCD parameters (Transmission parameters), the condition for the network to be 'symmetrical' is that the input and output ports are interchangeable. Mathematically, a network is symmetrical if and only if A = D.
For any reciprocal bilateral network, the relationship between the ABCD parameters is given by the condition:
AD - BC = 1
Given the values from the problem:
1. A = 3
2. B = 1
3. Since the network is symmetrical, A = D, therefore D = 3.
Substituting these values into the reciprocity condition:
(3 * 3) - (1 * C) = 1
9 - C = 1
C = 9 - 1
C = 8
Therefore, the value of parameter C is 8, which corresponds to option (d).
In measurement systems, which of the following static characteristic(s) is/are desirable?
A)
Accuracy
B)
Sensitivity
C)
Reproducibility
D)
All of these
In measurement systems, static characteristics are the properties of an instrument that remain constant when the input signal is constant or varies slowly.
1. Accuracy: This is the degree of closeness of a measured value to the true value of the quantity. High accuracy is always desirable to minimize measurement error.
2. Sensitivity: This is the ratio of the change in output (response) to the change in input (measurand). A higher sensitivity allows an instrument to detect smaller changes in the input, which is generally desirable for precision measurements.
3. Reproducibility: This refers to the ability of an instrument to provide the same output for the same input when measured by different operators, using different equipment, or under different environmental conditions. High reproducibility ensures the reliability and consistency of the data.
Since all three characteristics (Accuracy, Sensitivity, and Reproducibility) are positive attributes that improve the quality and reliability of a measurement system, the correct option is 'All of these'.
Maxwell’s divergence equation in case of static electric field is
A)
V.E=p
B)
VXE=p
C)
VE=2
D)
V.E=0
Maxwell's divergence equation for the electric field is based on Gauss's Law. In vector notation, Gauss's Law states that the divergence of the electric field (E) is equal to the volume charge density (ρ) divided by the permittivity of free space (ε₀). The mathematical expression is ∇ · E = ρ / ε₀. In the context of the provided options, 'V' represents the Nabla operator (∇), '.' represents the dot product (divergence), and 'p' represents the charge density (ρ). Therefore, V.E = p is the symbolic representation of the divergence of the electric field being proportional to the charge density.
Analysis of options:
- Option (a) V.E=p: Correct. It represents ∇ · E = ρ (Gauss's Law).
- Option (b) VXE=p: Incorrect. 'VXE' represents the curl (∇ × E). For a static electric field, the curl of E is zero (∇ × E = 0), not equal to charge density.
- Option (c) VE=2: Incorrect. This is mathematically meaningless in the context of Maxwell's equations.
- Option (d) V.E=0: Incorrect. This would only be true in a region where there is no charge density (charge-free space).
The magnitude of the induced e.m.f. in a conductor depends on the
A)
flux density of the magnetic field
B)
amount of flux cut
C)
amount of flux linkages
D)
rate of change of flux-linkages
According to Faraday's Law of Electromagnetic Induction, the magnitude of the induced electromotive force (e.m.f.) in a circuit is directly proportional to the rate of change of magnetic flux linkage with respect to time. Mathematically, this is expressed as: e = -N (dΦ/dt), where 'e' is the induced e.m.f., 'N' is the number of turns (linkages), and 'dΦ/dt' is the rate of change of magnetic flux.
Analysis of options:
- Option (a) Flux density (B) is a component of flux, but a constant flux density does not induce an e.m.f. unless there is relative motion or change over time.
- Option (b) The 'amount of flux cut' is a static value; induction requires the *rate* at which it is cut.
- Option (c) 'Amount of flux linkages' (NΦ) refers to the total flux, but again, a constant amount of linkage produces no e.m.f.
- Option (d) 'Rate of change of flux-linkages' is the precise definition provided by Faraday's Law, as the e.m.f. depends on how quickly the flux changes, not the absolute amount of flux present.