Problem 24 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Inductive battery chargers, which allow transfer of electrical power without the need for exposed electrical contacts, are commonly used in appliances that need to be safely immersed in water, such as electric toothbrushes. Consider the following simple model for the power transfer in an inductive charger (Fig. \(29-42\) ). Within the charger's plastic base, a primary coil of diameter \(d\) with \(n_{\mathrm{P}}\) turns per unit length is connected to a home's ac wall outlet so that a current \(I=I_{0} \sin (2 \pi f t)\) flows within it. When the toothbrush is seated on the base, an \(N\) -turn secondary coil inside the toothbrush has a diameter only slightly greater than \(d\) and is centered on the primary. Find an expression for the \(\mathrm{emf}\) induced in the secondary coil. [This induced emf recharges the battery.

Step-by-Step Solution

Answer

The induced emf in the secondary coil is \( \mathcal{E} = -N \mu_0 n_{\mathrm{P}} \pi^2 d^2 f I_0 \cos (2\pi f t) \).

Step 1: Understand the Problem

The problem requires us to find the expression for the electromotive force (emf) induced in the secondary coil of a toothbrush's charger system. This is a classic problem of electromagnetic induction where a varying current in a primary coil induces an emf in a secondary coil.

Step 2: Identify Given Quantities

We have a primary coil with diameter \(d\) and \(n_{\mathrm{P}}\) turns per unit length connected to an AC source with current \(I = I_0 \sin (2 \pi f t)\). The secondary coil has \(N\) turns and is coaxial with the primary coil.

Step 3: Use Faraday's Law of Induction

Faraday's law states that the induced emf (\(\mathcal{E}\)) in a coil is equal to the negative rate of change of magnetic flux through the coil. This can be expressed as: \( \mathcal{E} = -N \frac{d\Phi}{dt} \), where \(\Phi\) is the magnetic flux.

Step 4: Calculate the Magnetic Flux

The magnetic field \(B\) produced by the primary coil, which is a solenoid, is given by \(B = \mu_0 n_{\mathrm{P}} I \), where \(\mu_0\) is the permeability of free space. The flux through the secondary coil is \(\Phi = BA = \mu_0 n_{\mathrm{P}} I \cdot \frac{\pi d^2}{4}\).

Step 5: Differentiate the Magnetic Flux

Differentiate the expression for \(\Phi\) with respect to time \(t\): \( \frac{d\Phi}{dt} = \mu_0 n_{\mathrm{P}} \cdot \frac{\pi d^2}{4} \cdot \frac{dI}{dt} = \mu_0 n_{\mathrm{P}} \cdot \frac{\pi d^2}{4} \cdot 2\pi f I_0 \cos (2 \pi f t) \).

Step 6: Compute the Induced EMF

Substitute the derivative of the flux into Faraday's law to find the emf: \( \mathcal{E} = -N \frac{d\Phi}{dt} = -N \left( \mu_0 n_{\mathrm{P}} \cdot \frac{\pi d^2}{4} \cdot 2\pi f I_0 \cos (2\pi f t) \right) \). Simplify to get the final expression: \( \mathcal{E} = -N \mu_0 n_{\mathrm{P}} \pi^2 d^2 f I_0 \cos (2\pi f t) \).

Key Concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle in electromagnetism. It explains how an electric field (emf) is generated in a loop of wire when the magnetic flux through the loop changes.
This law is central to understanding how inductive charging works, including the charging of your electric toothbrush.
According to Faraday’s Law, the induced emf in any closed circuit is equal to the negative of the rate of change of magnetic flux embraced by the circuit. This can be mathematically described as:

• \( \mathcal{E} = -N \frac{d\Phi}{dt} \)

Here:

• \( \mathcal{E} \) is the electromotive force (emf)
• \( N \) is the number of turns in the coil
• \( \Phi \) is the magnetic flux
• \( \frac{d\Phi}{dt} \) is the rate of change of flux

This principle allows us to power devices without direct electrical contacts, thereby enhancing safety and efficiency.

Inductive Charging

Inductive charging is a fascinating application of electromagnetic induction, where power is transferred wirelessly from one coil (primary) to another (secondary).
This method eliminates the need for exposed electrical contacts, which is ideal for devices used in moist environments like electric toothbrushes.
In inductive charging, the primary coil is connected to a power source. An alternating current (AC) flows through this coil, creating a changing magnetic field around it.

• This magnetic field induces an emf in the secondary coil, situated on the receiving device (such as a toothbrush).
• The induced emf then charges the device's battery.

The efficiency of this process depends on:

• The alignment of the coils
• The distance between them
• The frequency of the AC current

Inductive charging is safe and reliable, which makes it a popular choice for many consumer applications.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, considering the strength and the extent of a magnetic field.
It plays a crucial role in determining the induced emf in a coil according to Faraday’s Law.
The magnetic flux \( \Phi \) through a given area \( A \) is calculated as:

• \( \Phi = B \cdot A \cdot \cos(\theta) \)

Where:

• \( B \) is the magnetic field strength
• \( A \) is the area through which the field lines pass
• \( \theta \) is the angle between the magnetic field and the normal (perpendicular) to the area

In the toothbrush charger example, magnetic flux changes as the AC current varies, which in turn induces emf that charges the device.

• The more tightly packed the field lines, the higher the magnetic flux.
• Flux is crucial for determining how effectively a secondary coil can pick up energy from a primary coil.

Electromotive Force (emf)

Electromotive force (emf) is the potential difference that drives the flow of electrons, creating an electric current.
Unlike its name suggests, emf is not actually a force but a potential difference, typically measured in volts.
In our problem, the emf is generated in the secondary coil due to the changing magnetic field produced by the primary coil.

• This induced emf is what ultimately powers and charges the electric toothbrush.
• The value of the induced emf depends on the rate of change of magnetic flux and the number of turns in the secondary coil.

The equation derived from Faraday’s law in this context is:

• \( \mathcal{E} = -N \cdot \frac{d\Phi}{dt} \)

Understanding how the emf is produced helps in designing more efficient wireless charging systems, enhancing their usability and reliability.