91Ó°ÊÓ

A circular conducting ring with radius \(r_{0}=0.0420 \mathrm{m}\) lies in the \(x y\) -plane in a region of uniform magnetic field \(\vec{B}=B_{0}\left[1-3\left(t / t_{0}\right)^{2}+\right.\) 2\(\left(t / t_{0}\right)^{3} \hat{\boldsymbol{k}} .\) In this expression, \(t_{0}=\) 0.0100 \(\mathrm{s}\) and is constant, \(t\) is time, \(\hat{\boldsymbol{k}}\) is the unit vector in the \(+z\) - direction, and \(B_{0}=0.0800 \mathrm{T}\) and is constant. At points \(a\) and \(b(\) Fig. \(\mathrm{P} 29.64)\) there is a small gap in the ring with wires leading to an external circuit of resistance \(R=12.0 \Omega .\) There is no magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux \(\Phi_{B}\) through the ring. (b) Determine the emf induced in the ring at time \(t=5.00 \times 10^{-3}\) s. What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through \(R\) at the time given in part (b) is only 3.00 mA. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time \(t=1.21 \times 10^{-2}\) s. What is the polarity of the emf? (e) Determine the time at which the current through \(R\) reverses its direction.

Step by step solution

01

Magnetic Flux Calculation

To find the magnetic flux \( \Phi_B \) through the ring, we start by calculating the magnetic field \( \vec{B} \) which is given as \( \vec{B} = B_0 \left[ 1 - 3\left( \frac{t}{t_0} \right)^2 + 2\left( \frac{t}{t_0} \right)^3 \right] \hat{\boldsymbol{k}} \). The flux \( \Phi_B \) is calculated using the formula:\[ \Phi_B = \int \vec{B} \cdot d\vec{A} = B_0 \left[ 1 - 3\left( \frac{t}{t_0} \right)^2 + 2\left( \frac{t}{t_0} \right)^3 \right] \pi r_0^2 \]where \( \vec{B} \) is perpendicular to the area vector \(d\vec{A}\).

02

Induced Emf Calculation (Part B)

According to Faraday's law, the induced emf is the negative rate of change of magnetic flux. Therefore, we differentiate the flux expression with respect to time:\[\varepsilon = -\frac{d\Phi_B}{dt} = -\pi r_0^2 B_0 \cdot \frac{d}{dt} \left( 1 - 3 \left( \frac{t}{t_0}\right)^2 + 2 \left( \frac{t}{t_0} \right)^3 \right)\]Simplifying, we find:\[\varepsilon = -\pi r_0^2 B_0 \left( -\frac{6t}{t_0^2} + \frac{6t^2}{t_0^3} \right)\]Substitute \( t = 5.00 \times 10^{-3} \) s to find \( \varepsilon \) and determine the polarity.

03

Current and Internal Resistance (Part C)

The current through resistance \( R \) is given, \( I = 3.00 \text{ mA} = 3.00 \times 10^{-3} \text{ A} \), at \( t = 5.00 \times 10^{-3} \text{ s} \). The total emf includes contributions from the internal resistance \( r \) of the ring:\[\varepsilon = I (R + r)\]Using the calculated \( \varepsilon \), rearrange to find \( r \).

04

Emf Calculation at Different Time (Part D)

Repeat the procedure from Step 2 for \( t = 1.21 \times 10^{-2} \text{ s} \). Replace \( t \) in the derivative of the magnetic field with respect to time to find \( \varepsilon \). Evaluate the polarity by checking the sign of \( \varepsilon \).

05

Time of Current Reversal (Part E)

The current reverses direction when the emf changes sign. Set the calculated \( \varepsilon \) expression equal to zero:\[0 = \pi r_0^2 B_0 \left(-\frac{6t}{t_0^2} + \frac{6t^2}{t_0^3}\right)\]Solve the resulting equation for \( t \) to find the time at which current reverses direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Flux

Magnetic flux is a fundamental concept in physics that describes the flow of the magnetic field through a given area. Imagine it as the amount of magnetic field 'passing through' a surface. In mathematical terms, magnetic flux \( \Phi_B \) through a surface is given by the integral of the magnetic field \( \vec{B} \) dot product with the area vector \( d\vec{A} \): \[ \Phi_B = \int \vec{B} \cdot d\vec{A} \] For a uniform magnetic field and a flat surface, this simplifies to:\[ \Phi_B = B \cdot A \cdot \cos(\theta) \] where \( B \) is the magnitude of the magnetic field, \( A \) is the area the field penetrates, and \( \theta \) is the angle between the magnetic field and the perpendicular (normal) to that surface. In our specific exercise, the magnetic field changes with time, which also means that the flux will change over time. Calculating this magnetic flux accurately is crucial for determining the energy and motion within electromagnetic systems.

Faraday's Law

Faraday's Law of electromagnetic induction is a key principle that explains how a change in magnetic flux leads to the generation of electromotive force (emf). It states that the induced emf \( \varepsilon \) is equal to the negative rate of change of magnetic flux through a circuit:\[ \varepsilon = -\frac{d\Phi_B}{dt} \] The negative sign in the equation, known as Lenz's law, indicates that the direction of the induced emf will oppose the change in flux causing it. This opposition is nature's way of maintaining stability and balance.

• When the magnetic flux through a circuit increases, the induced emf will generate a current that creates a magnetic field opposing the increase.
• Conversely, if the flux decreases, the induced emf will create a field to boost it back up.

In the problem we're examining, Faraday's Law is instrumental in finding the induced emf as the magnetic field changes with time, thus altering the flux through the circular ring.

Internal Resistance

Internal resistance is an inherent property of materials and circuits that resists the flow of electric current. This resistance can be found within the components of the circuit itself, such as a conducting ring or coil, and is different from external resistors added to the circuit. When current flows through a circuit, some energy is inevitably lost as heat within the wires and components, which is due to internal resistance.To find the internal resistance, we can rearrange the equation for the total effective emf and current in a circuit:\[ \varepsilon = I \cdot (R + r) \] where \( I \) is the current, \( R \) is the external resistance, and \( r \) is the internal resistance. By experimentally measuring the current and using known values for emf and external resistance, the internal resistance can be derived. Understanding internal resistance is vital for maximizing efficiency in circuits and ensuring they function properly without excess energy loss.

Induced EMF

Induced electromotive force (emf) occurs when a conductor experiences a change in magnetic flux, generating a voltage across it. This concept is a direct application of Faraday's Law and is essential in many electrical devices and systems. The induced emf can be calculated using the formula derived from differentiating the magnetic flux:\[ \varepsilon = -\frac{d\Phi_B}{dt} \] This means that any time the amount or direction of magnetic field lines passing through a conductor changes, an emf will be induced. In the setup given in the exercise, the emf is calculated based on the time-varying magnetic field that permeates the ring. By substituting time values into the derived expression for \( \varepsilon \), we can determine the magnitude and polarity of the resulting emf at different moments.

• The direction or polarity of this induced emf determines the direction of any resulting induced current.
• The calculations for induced emf are crucial for understanding how the circuit will behave over time and are fundamental in designing and analyzing transformers, electric motors, and generators.