Question

Part A

Which of the following statements are true regarding transformers?

Check all that apply.

Check all that apply.

	In a transformer, if the primary coil contains more loops than the secondary coil, then it is a step-up transformer.
	A transformer converts mechanical energy into electrical energy.
	A transformer is used to increase or decrease a direct current voltage.
	A transformer is used to increase or decrease an alternating current voltage.
	In a transformer, if the secondary coil contains more loops than the primary coil, then it is a step-up transformer.
	In a transformer, the power input is less than the power output.

Part B

A resistor, an inductor, and a switch are all connected in series to an ideal battery of constant terminal voltage. Suppose at first the switch is open, and then, at some initial time t = 0, it is closed. Which of the following statements are true?

Check all that apply.

Check all that apply.

	At the instant the switch is closed, the current reaches its steady-state value.
	The steady-state value of the current depends on the inductance of the inductor.
	At the instant the switch is closed, the current begins to increase at a rate that depends upon the value of the inductance of the inductor.
	The steady-state value of the current depends on the resistance of the resistor.

Part C

A resistor, an inductor, and a switch are all connected in series to an ideal battery of constant terminal voltage. What does the time constant for the circuit represent?

A resistor, an inductor, and a switch are all connected in series to an ideal battery of constant terminal voltage. What does the time constant for the circuit represent?

	The time constant represents the time required for the current to reach 25% of the maximum current.
	The time constant represents the time required for the current to reach 75% of the maximum current.
	The time constant represents the time required for the current to reach 37% of the maximum current.
	The time constant represents the time required for the current to reach 63% of the maximum current.
	The time constant represents the time required for the current to reach 50% of the maximum current.
	The time constant represents the time required for the current to reach 99% of the maximum current.

Part D

An inductor is connected in series to a fully charged capacitor. Which of the following statements are true?

Check all that apply.

Check all that apply.

	The stored electric field energy can be less than the stored magnetic field energy.
	As the capacitor is discharging, the current is increasing.
	The stored electric field energy can be equal to the stored magnetic field energy.
	As the capacitor is charging, the current is increasing.
	The stored electric field energy can be greater than the stored magnetic field energy.

Answer 1

Concepts and reason

The concept of the transformer properties, current in LR circuit and properties of LC circuit are used to solve this problem.

In the first part, use the properties of transformers to find out which of the option is correct for transformer.

In the second part, write down the expression of the current in LR circuit and find out the factors on which the current value depends.

In the third part, find out the value of the time constant for the LR circuit using the current expression.

In the last part, use the total energy stored in the LC circuit to find out what happens in the LC circuit when capacitor is fully charged.

Fundamentals

The expression of the transformer equation is given as follows:

$\frac{{{V_2}}}{{{V_1}}} = \frac{{{N_2}}}{{{N_1}}}$

Here, ${N_1}$ is the number of turns in primary coil, ${N_2}$ is the number of turns in the secondary coil, ${V_2}$ is the secondary voltage and ${V_1}$ is the primary voltage.

The expression of current in the LR circuit is given as follows:

$I = \frac{V}{R}\left( {1 - {e^{ - \left( {\frac{t}{{L/R}}} \right)}}} \right)$

Here, L is the inductance value, R is the resistance value, t is the time and V is the voltage value.

The total energy in the series LC circuit is given as follows:

$E = \frac{1}{2}C{V^2} + \frac{1}{2}L{I^2}$

Here, C is the capacitance and I is the current.

(A)

An electrical device that transfers electrical energy from one coil to another coil is defined as transformer.

When current is varying in the primary coil magnetic field is produced in the coil. According to the Faraday law of electromagnetic induction, the changing magnetic field induces voltage in the coil.

The transformer output power is always equal to the input power because in a transformer if the current value step-up then the transformer step-down the voltage value to make the output power same as input power.

The power in a transformer is given as:

$\begin{array}{l}\\{P_1} = {P_2}\\\\{I_1}{V_1} = {I_2}{V_2}\\\end{array}$

Here, ${P_1}$ is the power of primary coil and ${P_2}$ is the power of secondary coil.

Hence, in a transformer, if the primary coil contains more loops than the secondary coil, then it is a step-up transformer, a transformer converts mechanical energy into electrical energy, a transformer is used to increase or decrease a direct current voltage, and in a transformer, the power input is less than the power output are incorrect.

Transformers are used to change alternating voltages in the electrical power applications.

Hence, the option (4) i.e. A transformer is used to increase or decrease an alternating current voltage is correct.

In a step up transformer, the number of turns in secondary coil is more than the number of turns in primary coil.

The condition of the step-up transformer is given as:

${N_2} > {N_1}$

Hence, the option (5) i.e. In a transformer, if the secondary coil contains more loops than the primary coil, then it is a step-up transformer is correct.

(B)

The expression of current in the LR circuit is given as follows:

$I = \frac{V}{R}\left( {1 - {e^{ - \left( {\frac{t}{{L/R}}} \right)}}} \right)$

Here, L is the inductance value, R is the resistance value, t is the time and V is the voltage value.

From the above equation, we find out that the when the switch is closed, the current begins to increase exponentially and the increase rate depends upon the value of the inductance of the inductor.

The steady state value of the LR circuit is given as:

$\begin{array}{c}\\I = {I^\circ }\\\\ = \frac{V}{R}\\\end{array}$

The steady state value of the current depends on the resistance value of resistor not on the value of inductance.

Hence, at the instant, the switch is closed, the current reaches its steady-state value and the steady-state value of the current depends on the inductance of the inductor are not correct.

From the condition of the steady state value of the current, it is clear that it depends on the resistance value of resistor not on the value of inductance.

(C)

The expression of current in the LR circuit is given as follows:

$I = {I^\circ }\left( {1 - {e^{ - \left( {\frac{t}{\tau }} \right)}}} \right)$

Here, $\tau$ is the time constant.

At $t = \tau$ , the value of current is calculated as follows:

$\begin{array}{c}\\I\left( \tau \right) = {I^\circ }\left( {1 - \exp \left( { - 1} \right)} \right)\\\\ = {I^\circ }\left( {0.6321} \right)\\\end{array}$

The time required for the current to reach 63 % of the maximum current is defined as the time constant of the LR circuit.

(D)

The expression for the total energy in the series LC circuit is given as follows:

$E = \frac{1}{2}C{V^2} + \frac{1}{2}L{I^2}$

The conservation of energy states that the total energy in the series LC circuit remains conserved.

The current in the capacitor increases when the capacitor is discharging.

Hence, as the capacitor is charging, the current does not increase.

From the expression for the total energy, the energy stored in capacitor can be greater than, less than or equal to the energy stored in the inductor.

The capacitor is discharging means that charged stored in the capacitor decreases which implies that the potential value decreases.

Thus, the current increases to satisfy the law of conservation of energy.

In a series LC circuit, the stored electric field energy is equal to the stored magnetic field energy.

The energy stored in capacitor can be greater than, less than or equal to the energy stored in the inductor because the total energy remains conserved in a series LC circuit.

Ans: Part A.4

A transformer is used to increase or decrease an alternating current voltage.

Part A.5

In a transformer, if the secondary coil contains more loops than the primary coil, then it is a step-up transformer.

Part B.3

At the instant, the switch is closed the current begins to increase at a rate that depends upon the value of the inductance of the inductor.

Part B.4

The steady state value of the current depends on the resistance value of resistor.

Part C

The time constant represents the time required for the current to reach 63% of the maximum current.

Part D.1

The stored electric field energy can be less than the stored magnetic field energy.

Part D.2

As the capacitor is discharging, the current is increasing.

Part D.3

The stored electric field energy can be equal to the stored magnetic field energy.

Part D.5

The stored electric field energy can be greater than the stored magnetic field energy.