In a pair of adjacent coils, for a change of current in one of the coils from 0 A to 10 A in 0.25 s, the magnetic flux in the adjacent coil changes by 15 Wb. The mutual inductance of the coils is:
1.5 H
Understanding Mutual Inductance in Adjacent Coils
Mutual inductance is a property of two adjacent coils where a change in current in one coil induces an electromotive force (EMF) in the other coil. It quantifies how effectively a change in current in one coil can induce a voltage in a nearby coil through the magnetic flux linkage.
The relationship between the change in magnetic flux ($\Delta \Phi_2$) in the second coil and the change in current ($\Delta I_1$) in the first coil is given by the formula:
\(\Delta \Phi_2 = M \Delta I_1\)
where \(M\) is the mutual inductance between the two coils.
Calculating Mutual Inductance
We are given the following information:
- Initial current in the first coil: \(I_{1, \text{initial}} = 0 \text{ A}\)
- Final current in the first coil: \(I_{1, \text{final}} = 10 \text{ A}\)
- Change in current in the first coil: \(\Delta I_1 = I_{1, \text{final}} - I_{1, \text{initial}} = 10 \text{ A} - 0 \text{ A} = 10 \text{ A}\)
- Time taken for the change in current: \(\Delta t = 0.25 \text{ s}\) (Note: This information is not needed for calculating mutual inductance using the flux change).
- Change in magnetic flux in the adjacent coil: \(\Delta \Phi_2 = 15 \text{ Wb}\)
Using the formula \(\Delta \Phi_2 = M \Delta I_1\), we can solve for \(M\):
\(15 \text{ Wb} = M \times 10 \text{ A}\)
To find \(M\), we rearrange the equation:
\(M = \frac{\Delta \Phi_2}{\Delta I_1}\)
\(M = \frac{15 \text{ Wb}}{10 \text{ A}}\)
\(M = 1.5 \text{ H}\)
The mutual inductance of the coils is 1.5 H.
Result Summary
Based on the provided values, the mutual inductance between the adjacent coils is calculated to be 1.5 H.
Revision Table: Mutual Inductance Concepts
| Concept | Definition/Formula | Units |
|---|---|---|
| Mutual Inductance (\(M\)) | Relates change in flux in one coil to change in current in another: \(\Delta \Phi_2 = M \Delta I_1\) | Henry (H) |
| Magnetic Flux (\(\Phi\)) | Measure of the total magnetic field lines passing through a given area | Weber (Wb) |
| Current (\(I\)) | Flow of electric charge | Ampere (A) |
| Induced EMF (\(\mathcal{E}\)) | Voltage induced in a coil due to changing magnetic flux or current: \(\mathcal{E}_2 = -M \frac{dI_1}{dt}\) | Volt (V) |
Additional Information on Mutual Inductance
Mutual inductance depends on the geometry of the two coils (their size, shape, distance apart, and relative orientation) and the properties of the core material between them (if any). It is a reciprocal property; the mutual inductance from coil 1 to coil 2 is the same as the mutual inductance from coil 2 to coil 1 (\(M_{12} = M_{21} = M\)).
Mutual inductance is important in many electrical components and circuits, including transformers, induction motors, and some types of sensors. In transformers, the principle of mutual inductance is used to transfer energy between coils with different numbers of turns to step up or step down voltage.
The unit of mutual inductance, the Henry (H), is defined as the mutual inductance between two circuits when a current change of one ampere per second in one circuit induces an electromotive force of one volt in the other circuit. Alternatively, as used in this problem, it can be defined based on flux linkage: one Henry is the mutual inductance when a change of 1 Ampere in one coil causes a change of 1 Weber of magnetic flux in the other coil.
1.5 H
Using the formula: \( M = \frac{\Delta \Phi}{\Delta I} \)
\( \Delta \Phi = 15 \, \text{Wb}, \quad \Delta I = 10 \, \text{A} \)
\( M = \frac{15}{10} = 1.5 \, \text{H} \)
Answer:
1.5 H
1.5 H
Given:
- Change in current (ΔI) = 10 A - 0 A = 10 A
- Change in flux (ΔΦ) = 15 Wb
Step 1: Formula for Mutual Inductance (M)
\[ M = \frac{\Delta \Phi}{\Delta I} \]
Step 2: Plug in the values
\[ M = \frac{15\ \text{Wb}}{10\ \text{A}} = 1.5\ \text{H} \]
Note: The time interval (0.25 s) is not needed for this calculation as we're given the total flux change, not the rate of change of flux.
Final Answer:
The mutual inductance of the coils is \[ \boxed{1.5\ \text{H}} \].
Important Questions from Electromagnetic Induction
A 50 Hz AC current of crest value 1 A flows through the primary of a transformer. If the mutual inductance between the primary and secondary is 0.5 H, the crest voltage induced in the secondary is:
A long solenoid of diameter 0.1 m has 2 × 104 turns per meter. At the center of the solenoid, a coil of 100 turns and radius 0.01 m is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to 0 A from 4 A in 0.05 s. If the resistance of the coil is 10π² Ω, then the total charge flowing through the coil during this time is:
Lower half of a convex lens is made opaque. Which of the following statements describes the image of the object placed in front of the lens?
A transformer has an efficiency of 80%. It works at 3 kW and 120 V. If the secondary voltage is 240 V, what will be the secondary current?
In an AC generator when the plane of the armature is perpendicular to the magnetic field, what will the magnitude of the magnetic flux passing through the coil and the emf induced in the coil be?
