6.4: Inductance and Transformers

Learning Objectives

• Determine the self-inductance of a coil and relate emf to current through the concept of self-inductance.
• Determine the mutual inductance of two coils and relate emf to current through the concept of mutual inductance.
• Derive step-up and step-down transformer equations.

Model

Explore

1. Rank the four coils from greatest to least for the following properties. If any are equal, place an equal sign between them.
   • Length
   • Number of Rings
   • Magnetic Field Strength
   • Total Area
2. Write an expression (and simplify if possible) for the magnetic field created by each coil using the given values or variables.

   \[ B_A=\qquad B_B=\qquad B_C=\qquad B_D= \]

Critical Thinking & Concept Invention

1. Rank the magnetic flux through the four coils from greatest to least.
2. Analyze the coils.
   • Write an expression (and simplify if possible) for the magnetic flux through each coil based on the magnetic fields they create.
   • As a group, identify what variable in the magnetic flux expression might vary with time depending on the circuit. Rewrite that variable with explicit time dependence (for example, \(V(t)\)) and make it the last quantity in the expression.
   • Identify the induced emf in each circuit. Recall:

   \[ \varepsilon=-\frac{d\Phi}{dt} \]

   • Recall that for inductors:

   \[ \Delta V_L=-L\frac{dI}{dt} \]

   • Circle the constants in front of \(\dfrac{dI}{dt}\).
   • Based on your expression for \(\varepsilon\), identify \(L\) for each coil.

   \[ \Phi_A=\qquad \Phi_B=\qquad \Phi_C=\qquad \Phi_D= \]

   \[ \Phi_A=\qquad \Phi_B=\qquad \Phi_C=\qquad \Phi_D= \]

   \[ \varepsilon_A=\qquad \varepsilon_B=\qquad \varepsilon_C=\qquad \varepsilon_D= \]

   \[ L_A=\qquad L_B=\qquad L_C=\qquad L_D= \]

Summary

Faraday’s Law is the fundamental law upon which Kirchhoff’s Voltage Rule is based. It explains all voltages and emfs in a circuit.

\[ \Gamma_E=-\frac{d\Phi_B}{dt} \]

\[ \oint \vec{E}\cdot d\vec{l}+\frac{d\Phi_B}{dt}=0 \]

\[ \oint \vec{E}\cdot d\vec{l}=-\sum \Delta V \]

\[ -\sum \Delta V+L\frac{dI}{dt}=0 \]

\[ \sum \Delta V-L\frac{dI}{dt}=0 \]

The \(\sum \Delta V\) terms represent changes to electric potential and include voltages from resistors, capacitors, batteries, and other voltage sources. These voltages arise from electric fields created by charges.

Previously, the relationship

\[ \Delta V_L=-L\frac{dI}{dt} \]

was introduced. This quantity is not actually electric potential because it is produced by changing magnetic fields rather than charges, but it behaves similarly in circuit analysis. The "voltage" associated with an inductor is identified through Faraday’s Law.

Apply

A solenoid 50 cm long is wound with 500 turns of wire. The cross-sectional area of the coil is \(2.0\ \text{cm}^2\).

• Determine the self-inductance of the solenoid.
• Draw a picture and label the quantities.

Model

Two coils are aligned on their common axis ("coaxial cable"). Each coil has the same total number of loops. Scenarios A and B have the same current, but differ in which coil carries the current.

Explore

1. Compare the two coils using \(>\), \(<\), or \(=\).
   • \(A_1\) ___ \(A_2\)
   • \(L_1\) ___ \(L_2\)
   • \(n_1\) ___ \(n_2\)
2. Compare the magnetic field and flux for each scenario.

   \[ B_1\qquad B_2 \]

3. Compare the flux in each solenoid for Scenario A.

   \[ \Phi_1\qquad \Phi_2 \]

4. Compare the flux in each solenoid for Scenario B.

   \[ \Phi_1\qquad \Phi_2 \]

Critical Thinking & Concept Invention

1. Compare the effects of one coil on another.
   • Identify the flux in each coil caused by the magnetic field of the other coil.
   • Rewrite that expression as a relationship to the current in the other coil.
   • When two coils affect each other, their relationship is \(\Phi_1=M_1I_2(t)\). Identify \(M_1\) and \(M_2\).
   • Compare \(M_1\) and \(M_2\) using \(>\), \(<\), or \(=\).
   • Calculate the emf induced in each coil by the other coil. Express your answers in terms of \(M\). If variables are identical, subscripts may be omitted.

   \[ \Phi_1=\qquad \Phi_2= \]

   \[ \Phi_1=I_2(t)\qquad \Phi_2=I_1(t) \]

   \[ M_1=\qquad M_2= \]

   \[ M_1\qquad M_2 \]

   \[ \varepsilon_1=\qquad \varepsilon_2= \]

Apply

A solenoid with \(N_1\) turns has length \(L_1\) and radius \(R_1\), while a second smaller solenoid with \(N_2\) turns has length \(L_2\) and radius \(R_2\). The smaller solenoid is placed completely inside the larger solenoid so that their long axes coincide.

• Draw a picture and label the quantities.
• Determine the mutual inductance of the two solenoids.

Model

Two solenoids with different loop densities are shown. Both have the same length and are assumed to be close enough (or interwoven) to neglect edge effects. The lower solenoid (A) carries current \(I_A\), producing the magnetic field shown. As illustrated, \(N_A<N_B\), although the drawing is not necessarily to scale.

Explore

1. Compare the magnetic field in each.

   \[ B_A=B_B \qquad\text{OR}\qquad > \]

2. Compare the magnetic flux per ring in each.

   \[ \frac{\Phi_1}{N_A}=\frac{\Phi_2}{N_B} \qquad\text{OR}\qquad > \]

3. Compare the total magnetic flux through each.

   \[ \Phi_A=\Phi_B \qquad\text{OR}\qquad > \]

Critical Thinking & Concept Invention

1. Compare quantities for each solenoid. Write your derivation in the space provided.
   • Construct an equation relating the total magnetic flux in each solenoid.
   • Take the time derivative of that relationship, then compare the emf in each.
   • Identify the circumstances under which \(\varepsilon_A>\varepsilon_B\).
   • Identify the circumstances under which \(\varepsilon_A<\varepsilon_B\).

Apply

A transformer is made from a solenoidal coil with 26 turns wound tightly around another coil with 330 turns. The inner solenoid is 20.0 cm long and has a diameter of 2.5 cm.

• Determine the mutual inductance of this transformer.
• If the 330-turn solenoid has a voltage peak of 120 V, determine the induced emf in the other coil.
• Determine the required rate of change of current, \(\dfrac{dI}{dt}\), in the 330-turn solenoid that would create these voltages.