Self Inductance: Videos & Practice Problems

Self-inductance is the ability of a current-carrying wire to induce an electromotive force (emf) on itself by changing its magnetic flux. The relationship is expressed as $\Phi =n\alpha I$, where $L$ is the self-inductance constant. The self-induced emf can also be calculated using $\mathrm{emf}=-L∆I/∆t$. This property depends on the coil's shape and the number of turns, making it a fundamental aspect of electromagnetism.

Self Inductance

Self-inductance is a fundamental concept in electromagnetism that describes how a current-carrying wire can induce an electromotive force (EMF) on itself due to changes in its magnetic flux. When a current flows through a coil of wire, it generates a magnetic field, which in turn creates magnetic flux through the coil. The magnetic flux (\( \Phi_B \)) is defined as the product of the magnetic field (\( B \)) and the area (\( A \)) through which the field lines pass, adjusted for the angle (\( \theta \)) between the field and the area vector: \( \Phi_B = B \cdot A \cdot \cos(\theta) \). In the case of a single loop where the magnetic field and area vector are aligned, \( \cos(\theta) \) equals 1, simplifying the equation to \( \Phi_B = B \cdot A \).

The total magnetic flux through the coil depends on the number of turns (\( n \)), the magnetic field, and the area. Since the magnetic field is influenced by the current (\( I \)), there exists a proportional relationship between the magnetic flux and the current: \( \Phi_B \propto I \). This relationship can be expressed mathematically as:

\[ \Phi_B = L \cdot I \]

where \( L \) is the self-inductance of the coil, measured in henrys (H). The self-inductance represents the coil's ability to induce an EMF on itself as the current changes. The units of self-inductance can also be expressed as webers per ampere (Wb/A).

To calculate the self-inductance, the formula can be rearranged to:

\[ L = \frac{\Phi_B}{I} \]

Self-inductance is a property that depends solely on the physical characteristics of the coil, such as the number of turns and its shape, rather than the current itself. This means that when calculating self-inductance, the current will cancel out, leaving \( L \) as a constant characteristic of the coil.

Additionally, the self-induced EMF can be expressed using Faraday's law, which relates the induced EMF to the rate of change of magnetic flux. This can also be represented in terms of self-inductance as:

\[ \text{EMF} = -L \frac{\Delta I}{\Delta t} \]

In this equation, \( \Delta I/\Delta t \) represents the rate of change of current over time. This relationship allows for the calculation of the induced EMF based on how quickly the current is changing.

For a single current-carrying loop, the self-inductance can be derived by substituting the expression for magnetic flux into the self-inductance formula. The magnetic field for a loop of wire is given by:

\[ B = \frac{\mu_0 I}{2r} \]

where \( \mu_0 \) is the permeability of free space and \( r \) is the radius of the loop. The area of the loop is \( A = \pi r^2 \). By substituting these values into the flux equation and simplifying, the self-inductance can be expressed as:

\[ L = \frac{\mu_0 \pi r}{2} \]

This result shows that the self-inductance of a single loop is dependent on the physical dimensions of the loop and the permeability of the medium, reinforcing the idea that self-inductance is a characteristic property of the coil itself.

A single loop of wire with a current of 0.3A produces a flux of 0.005 Wb. If the self-induced EMF on this loop is 10 mV, how quickly must the current be changing?

• A. 1.67×10⁻⁴ A/s
• B. 0.60 A/s
• C. 1.67 A/s
• D. 60 A/s

Answer: (B)

Self-Inductance of a Toroid

In this example, we explore the calculation of the self-inductance of a toroidal solenoid and the induced electromotive force (EMF) when the current changes. A toroidal solenoid can be visualized as a series of coils wrapped around a circular path, resembling a donut shape. To determine the self-inductance, we utilize the formula:

$$ L = \frac{N \Phi}{I} $$

where \( L \) is the self-inductance, \( N \) is the number of turns, \( \Phi \) is the magnetic flux, and \( I \) is the current. In this case, we have 500 turns, a cross-sectional area of \( 6.25 \, \text{cm}^2 \) (which converts to \( 6.25 \times 10^{-4} \, \text{m}^2 \)), and a mean radius of \( 4 \, \text{cm} \) (or \( 0.04 \, \text{m} \)). The magnetic flux \( \Phi \) can be expressed as:

$$ \Phi = B \cdot A $$

where \( B \) is the magnetic field inside the toroid, given by:

$$ B = \frac{\mu_0 N I}{2 \pi r} $$

Here, \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{H/m} \). By substituting the expression for \( B \) into the flux equation and then into the self-inductance formula, we find:

$$ L = \frac{N^2 \mu_0 A}{2 \pi r} $$

Plugging in the values, we calculate the self-inductance \( L \) to be approximately \( 7.81 \times 10^{-4} \, \text{H} \).

Next, we examine the induced EMF when the current decreases from \( 5 \, \text{A} \) to \( 2 \, \text{A} \), resulting in a change in current \( \Delta I = -3 \, \text{A} \) over a time interval of \( 3 \, \text{ms} \) (or \( 0.003 \, \text{s} \)). The induced EMF \( \epsilon \) can be calculated using the formula:

$$ \epsilon = -L \frac{\Delta I}{\Delta t} $$

Substituting the known values, we find:

$$ \epsilon = -7.81 \times 10^{-4} \cdot \frac{-3}{0.003} $$

This results in an induced EMF of approximately \( 0.78 \, \text{V} \). Thus, we have successfully calculated both the self-inductance of the toroidal solenoid and the induced EMF due to a changing current.

Here’s what students ask on this topic:

What is self-inductance and how is it calculated?

Self-inductance is the ability of a current-carrying wire to induce an electromotive force (emf) on itself by changing its magnetic flux. It is represented by the symbol L. The self-inductance of a coil can be calculated using the formula:

$L=\frac{n\Phi }{I}$

where n is the number of turns, Φ is the magnetic flux, and I is the current. The unit of self-inductance is the henry (H), which is equivalent to a Weber per ampere (Wb/A).

How does the shape of a coil affect its self-inductance?

The shape of a coil significantly affects its self-inductance. The self-inductance depends on the coil's geometry, including its length, radius, and the number of turns. For example, a solenoid (a coil wound into a tightly packed helix) has a different self-inductance compared to a single loop of wire. The formula for the self-inductance of a solenoid is:

$L=\frac{\mu n{N}^{2}A}{l}$

where μ is the permeability of the core material, N is the number of turns, A is the cross-sectional area, and l is the length of the solenoid.

What is the relationship between self-inductance and emf?

The relationship between self-inductance (L) and electromotive force (emf) is given by Faraday's law of induction. The self-induced emf in a coil is proportional to the rate of change of current through the coil. This relationship is expressed by the formula:

$\mathrm{emf}=-L\frac{∆I}{∆}$

where ΔI/Δt is the rate of change of current. The negative sign indicates that the induced emf opposes the change in current, according to Lenz's law.

What are the units of self-inductance and how are they derived?

The unit of self-inductance is the henry (H). It is derived from the relationship between magnetic flux (Φ), current (I), and self-inductance (L). The formula for self-inductance is:

$L=\frac{n\Phi }{I}$

where n is the number of turns, Φ is the magnetic flux in Webers (Wb), and I is the current in amperes (A). Therefore, the unit of self-inductance (L) is Weber per ampere (Wb/A), which is defined as one henry (H).

How does the number of turns in a coil affect its self-inductance?

The number of turns (n) in a coil directly affects its self-inductance (L). The self-inductance is proportional to the square of the number of turns. This relationship is given by the formula:

$L=\frac{\mu n{N}^{2}A}{l}$

where μ is the permeability of the core material, N is the number of turns, A is the cross-sectional area, and l is the length of the coil. Increasing the number of turns increases the self-inductance, making the coil more effective at inducing emf.