Alternating Current

Step 1: Understanding the Concept:
Resonance in a series L-C-R circuit is a condition where the inductive reactance and capacitive reactance cancel each other out, leading to minimum impedance and maximum current.
Step 2: Key Formula or Approach:
The impedance ( ) of a series L-C-R circuit is given by:
where is the inductive reactance and is the capacitive reactance.
The phase difference ( ) between voltage and current is given by:
Step 3: Detailed Explanation:
The resonance condition occurs when the inductive reactance equals the capacitive reactance:
Substituting this condition into the formula for the phase difference:
If , then the phase angle must be zero.
When the phase difference is zero, the applied voltage and the current are in the same phase. This means the circuit behaves as a purely resistive circuit at resonance.
Step 4: Final Answer:
At resonance, the phase difference between the applied voltage and current is zero. Therefore, option (D) is correct.

Step 1: Understanding the Concept:
A transformer is a device that transfers electrical energy from one circuit to another through electromagnetic induction. A step-up transformer increases the AC voltage from the primary coil to the secondary coil.
Step 2: Key Formula or Approach:
For an ideal transformer (assuming 100% efficiency and no energy loss), the power input to the primary coil is equal to the power output from the secondary coil.
Power , where is voltage and is current.
Therefore, for an ideal transformer:
where denotes the primary coil and denotes the secondary coil.
Step 3: Detailed Explanation:
In a step-up transformer, the secondary voltage is greater than the primary voltage:
Using the power conservation equation , we can write the ratio of the currents as:
Since , the ratio is less than 1.
This implies that .
So, in a step-up transformer, while the voltage is increased, the current is decreased proportionally.
Step 4: Final Answer:
The value of the current in the secondary coil of a step-up transformer is less than the current in the primary coil. Option (B) is correct.

Step 1: Understanding the Concept:
The root mean square (r.m.s) value of an alternating current (AC) represents the effective DC value that would deliver the same average power to a resistor. For a sinusoidal current, it is directly related to its peak (maximum) value.
Step 2: Key Formula or Approach:
The general equation for a sinusoidal AC is given by: where is the peak amplitude or maximum current.
The r.m.s value ( ) is related to the peak current by the formula: Step 3: Detailed Explanation:
The given equation for the current is .
By comparing this with the standard form , we can identify the peak current: Now, we apply the formula for the r.m.s value: To simplify this expression, we rationalize the denominator by multiplying the numerator and denominator by : Step 4: Final Answer:
The calculated r.m.s value is A, which is approximately A. This value does not match any of the given options. There appears to be an error in the question or the options. In such cases in an exam, it is common that there was a typo in the question. For instance, if the equation had been , then would be A, matching option (B). Given the choices, it's highly probable this was the intended question. However, based strictly on the question provided, none of the options is correct.

Step 1: Understanding the Concept:
Impedance ( ) is the total opposition that a circuit presents to the flow of alternating current. In a series L-R circuit, it arises from two sources: the resistance ( ) of the resistor and the inductive reactance ( ) of the inductor. These two quantities do not add arithmetically because the voltage across the resistor is in phase with the current, while the voltage across the inductor leads the current by 90 degrees.
Step 2: Key Formula or Approach:
The resistance is .
The inductive reactance is given by , where is the angular frequency of the AC supply.
Since and are out of phase by 90°, they are combined as vectors (or phasors) using the Pythagorean theorem. This can be visualized using an impedance triangle where R is the adjacent side, is the opposite side, and Z is the hypotenuse. Step 3: Detailed Explanation:
Substitute the expression for inductive reactance ( ) into the general formula for impedance: This simplifies to: Step 4: Final Answer:
Comparing the derived expression with the given options, we see that option (D) correctly represents the impedance of a series L-R circuit.

Step 1: Understanding the Concept:
Consider a series circuit containing an inductor (L), a capacitor (C), and a resistor (R) connected to an alternating voltage source . The circuit offers opposition to the current flow, which is called impedance ( ). Resonance is a special condition in this circuit where the current becomes maximum.
Step 2: Key Formula or Approach:
The inductive reactance is , which leads the current by a phase of .
The capacitive reactance is , which lags behind the current by a phase of .
The total impedance ( ) of the series L-C-R circuit is given by the phasor diagram analysis:
The peak current ( ) in the circuit is given by Ohm's law for AC circuits:
Step 3: Deriving the Condition for Resonance:
Electrical resonance occurs when the current in the circuit reaches its maximum possible value. From the equation for , it is clear that for a given and R, the current will be maximum when the impedance is minimum.
The impedance will be minimum when the term is minimum, which is zero.
This is the condition for resonance in a series L-C-R circuit. At resonance, the inductive reactance is exactly equal to the capacitive reactance. The impedance becomes purely resistive, , and the current is in phase with the voltage.
Step 4: Finding the Expression for Resonant Frequency:
Let the angular frequency at which resonance occurs be the resonant angular frequency, denoted by .
Using the resonance condition:
Rearranging the terms to solve for :
This is the expression for the resonant angular frequency.
The ordinary frequency ( ) is related to the angular frequency by .
This is the expression for the resonant frequency. It is the natural frequency of oscillation of the circuit.

To convert a galvanometer into an ammeter, a low-resistance resistor called a shunt is connected in parallel with the galvanometer coil. This arrangement allows most of the current to bypass the galvanometer, preventing it from being damaged by large currents. The shunt resistor is chosen such that only a small, safe fraction of the total current passes through the galvanometer, while the remainder flows through the shunt. This enables the device to measure much higher currents than the galvanometer’s full-scale deflection current. By carefully selecting the value of , the ammeter can provide an accurate measurement of the total current , which is the sum of the currents through the galvanometer and the shunt : Thus, the galvanometer with a shunt resistor acts as an ammeter capable of measuring large currents safely.

In a moving coil galvanometer, the deflection of the coil is directly proportional to the current passing through it. This is because the torque experienced by the coil due to the magnetic field is proportional to the current, and the restoring torque of the suspension wire balances it at equilibrium. Mathematically, this relationship can be expressed as: or equivalently, where is a constant of proportionality dependent on the galvanometer's construction. This linear relationship allows the galvanometer to measure current by observing the angular deflection of its coil.

The quantity represents the speed of light in CGS units, which is approximately: Given that option (A) is , which equals , this appears to be a typo or error in the units. The correct value for the speed of light in CGS units is closer to , thus option (B) is intended to represent this value.

The magnetic force on a charged particle is given by: If the velocity is parallel to the magnetic field , then the cross product is zero: Thus, the force acting on the electron is zero.

Step 1: When the current to be measured is very large, a low-resistance device called a shunt is connected in parallel with the ammeter. The shunt provides an alternative path for the current, thereby protecting the sensitive coil of the ammeter from excessive current.
Step 2: Since the shunt has a very low resistance compared to the internal resistance of the ammeter, most of the current flows through the shunt, and only a small, known fraction of the total current flows through the meter. This small current is within the safe operating limits of the ammeter.
Step 3: This arrangement prevents damage to the ammeter by avoiding overload and allows the device to measure high currents accurately. The total current can be calculated using the known resistance of the shunt and the reading on the ammeter, making it a practical method for extending the current range of an ammeter.

Step 1: In an alternating current (AC) circuit, the total current can be resolved into two distinct components based on its phase relationship with the applied voltage: - The active current (or real component) which is in phase with the voltage. This component is responsible for performing real work, such as producing heat, light, or mechanical energy. - The wattless current (or reactive component), which is out of phase with the voltage by . This component does not perform any net work over a full cycle; instead, it simply transfers energy back and forth between the source and the reactive components (inductors and capacitors) of the circuit.
Step 2: The wattless current primarily arises in circuits with purely inductive or purely capacitive loads, where the phase difference between voltage and current is exactly . In such cases, the current either leads or lags the voltage by a quarter cycle, resulting in zero average power consumption.
Step 3: The power consumed in an AC circuit is given by the expression: where is the RMS voltage, is the RMS current, and is the phase angle between voltage and current. For wattless current, , hence: Therefore, the power associated with wattless current is zero, confirming that it does not contribute to any real energy transfer in the circuit.

Step 1: Write down the place values for each binary digit, starting from the rightmost digit (least significant bit). The place values are the powers of 2, increasing from right to left: For example, for a 5-bit binary number like , the place values would be:
Step 2: Multiply each binary digit (bit) by its corresponding power of 2. Consider the binary number :
Step 3: Add all the products obtained in Step 2 to get the decimal equivalent: Therefore, the binary number is equal to in decimal.

Step 1: UNIVAC (Universal Automatic Computer) was one of the earliest commercial computers developed in the United States in the early 1950s. It was a pioneering machine in the history of computing, known for its remarkable speed, accuracy, and reliability in processing large volumes of data. UNIVAC played a significant role in applications such as the U.S. Census and business data processing, marking the beginning of the computer age in commercial environments.
Step 2: Fax machines (short for facsimile machines) are devices that allow documents to be transmitted electronically over telephone lines. The machine scans a physical document and converts it into electronic signals, which are then sent to a receiving fax machine. The receiving machine reconstructs the electronic data into a printed copy of the original document. This process enables rapid sharing of printed documents across long distances, making fax machines an essential tool for communication, especially before the widespread adoption of email and digital document transfer.

The energy stored in a condenser (capacitor) is given by where is the capacitance and is the potential difference. Initial energy: Final energy: Increase in energy:

Ohmic Resistance:
A resistance is called ohmic if it obeys Ohm's law, which states that the current flowing through a conductor is directly proportional to the applied voltage across it, provided the temperature remains constant. Mathematically, this can be expressed as: where is the resistance and remains constant for ohmic materials.
The graphical representation of voltage versus current (the characteristic) for an ohmic resistor is a straight line passing through the origin, indicating a constant resistance.
Example: Common conductors such as copper wires and metallic resistors exhibit ohmic behavior under normal conditions.
Non-Ohmic Resistance:
A resistance is termed non-ohmic if it does not obey Ohm's law, meaning the current flowing through the device is not directly proportional to the applied voltage. In such cases, the resistance changes with voltage or current, often due to changes in temperature or other physical properties.
The characteristic curve for a non-ohmic device is nonlinear, indicating that the resistance varies with applied voltage or current.
Example: Devices like diodes and filament bulbs are non-ohmic. For instance, in a filament bulb, the filament heats up as current passes through it, increasing its resistance and causing a nonlinear curve.

Step 1: In a cyclotron, charged particles (such as protons or ions) are injected at the center between two hollow semi-circular electrodes called \emph{dees}. These particles are accelerated each time they cross the gap between the dees by an alternating electric field.
Step 2: A strong magnetic field , applied perpendicular to the plane of the dees, forces the charged particles to move in a circular path due to the Lorentz force. As the particles gain energy and speed with each acceleration, the radius of their circular trajectory increases.
Step 3: The frequency of revolution of the charged particles in the magnetic field is given by: where is the charge of the particle, is the magnetic field strength, and is the mass of the particle. This frequency, known as the cyclotron frequency, remains constant for non-relativistic speeds.
Step 4: However, for very light particles like electrons, when their speeds approach a significant fraction of the speed of light, relativistic effects become important. The effective mass of the particle increases, which reduces the frequency of revolution. This causes the particle to fall out of sync with the alternating electric field, leading to synchronization failure and limiting the cyclotron's ability to accelerate electrons efficiently.

A hot wire ammeter measures the root mean square (RMS) value of an alternating current (AC) because it operates based on the heating effect produced by the current passing through a wire. The heating effect (Joule heating) is proportional to the square of the current , i.e., For an alternating current, the instantaneous current varies with time, but the wire heats up according to the average power dissipated, which depends on the mean of the square of the current. Therefore, the hot wire ammeter effectively responds to the RMS value of the AC, which represents the equivalent direct current (DC) value that would produce the same heating effect in the wire.

An alternating current (AC) varies sinusoidally with time, completing both positive and negative half cycles in each full cycle. During the positive half cycle, the current flows in one direction, and during the negative half cycle, it flows in the opposite direction. When calculating the average (mean) value of the current over one full cycle, the positive and negative contributions cancel each other out because they are equal in magnitude but opposite in direction. Mathematically, this results in: Thus, the mean value of an AC over a complete cycle is zero.

In an alternating current (AC) circuit, the power factor is defined as the cosine of the phase difference between the voltage (electromotive force) and the current. It represents the fraction of the total power that is effectively used to do useful work. Mathematically, where is the angle by which the current either leads or lags the voltage. A power factor of 1 indicates that voltage and current are in phase, and all the power is effectively utilized, whereas a lower power factor indicates the presence of reactive components causing energy to oscillate back and forth without being consumed.

In an alternating current (AC) circuit, power loss, which is the energy dissipated as heat, occurs only in resistive components because resistors convert electrical energy into thermal energy continuously. Inductors and capacitors, on the other hand, store energy temporarily in their magnetic and electric fields respectively, and release it back to the circuit without dissipating it as heat. As a result, ideal inductors and capacitors do not cause any power loss. The average power dissipated in these reactive components over a full AC cycle is zero. Therefore, power loss in an AC circuit is solely due to the resistive parts.

The general form of AC current is . Given: , so

Electroplating requires the use of direct current (DC) because DC provides a unidirectional flow of electric current. This steady flow causes metal ions in the electrolyte to move consistently toward the cathode (negative electrode), where they gain electrons and deposit as a thin, uniform metal layer. If alternating current (AC) were used instead, the direction of current would continuously reverse, causing metal ions to deposit and then dissolve alternately. This would prevent uniform plating and could lead to inefficient or no metal deposition. Thus, DC is essential for controlled and effective electroplating.

Step 1: Formula for Root Mean Square (RMS) value.
The relationship between the peak value ( ) and the root mean square (RMS) value ( ) for alternating current is given by: where is the peak current.
Step 2: Calculate the RMS value.
Substitute the peak value into the formula:
Step 3: Analyze options.
- (A) 5 A: Incorrect. This is not the RMS value.
- (B) 7.07 A: Correct. This is the RMS value for a peak value of 10 A.
- (C) 10 A: Incorrect. The peak value is given, not the RMS value.
- (D) 14.14 A: Incorrect. This is the value for the peak value itself.
Step 4: Conclusion.
The RMS value of the current is 7.07 A.