Problem 47 - University Physics with Modern Physics

Question

A circular wire loop of radius \(a\) and resistance \(R\) initially has a magnetic flux through it due to an external magnetic field. The extermal field then decreases to zero. A current is induced in the loop while the external field is changing; however, this current does not stop at the instant that the external field stops changing. The reason is that the current itself generates a magnetic field, which gives rise to a flux through the loop. If the current changes, the flux through the loop changes as well, and an induced emf appears in the loop to oppose the change. (a) The magnetic field at the center of the loop of radius a produced by a current i in the loop is given by \(B=\mu_{0} i / 2 a\) . If we use the crode approximation that the field has this same value at all points within the loop, what is the flux of this field through the loop? (b) By using Faraday's law, Eq. \((29.3),\) and the relationship \(\mathcal{E}=i R,\) show that after the external field has stopped changing, the current in the loop obeys the differential equation $$ \frac{d i}{d t}=-\left(\frac{2 R}{\pi \mu_{0} a}\right) i $$ (c) If the current has the value \(i_{0}\) at \(t=0\) , the instant that the external field stops changing. snive the equation in part \((b)\) to find \(i\) as a function of time for \(t>0 .\) (Hint: In Section 26.4 we encountered a similar differential equation, Eq. \((26.15),\) for the quantity \(q .\) This equation for \(i\) may be solved in the same way. (d) If the loop has radius \(a=50 \mathrm{cm}\) and resistance \(R=0.10 \Omega,\) how long after the external field stops changing will the current be equal to 0.010 \(\mathrm{o}\) (that is, \(\frac{1}{100}\) of its initial value)? (e) In solving the examples in this chapter, we ignored the effects described in this problem. Explain why this is a good approximation.

Step-by-Step Solution

Answer

(a): \( \frac{\mu_0 \pi a i}{2} \); (b): \( \frac{di}{dt} = -\frac{2R}{\pi \mu_0 a} i \); (c): \( i(t) = i_0 e^{-\frac{2R}{\mu_0 \pi a} t} \); (d): \( t = \frac{\mu_0 \pi a}{2R} \ln(100) \).

Step 1: Compute the Flux for Part (a)

The magnetic field at the center produced by a current \(i\) in the loop is \(B = \frac{\mu_0 i}{2a}\). The crude approximation suggests this field is constant across the loop. The area of the loop is \( A = \pi a^2 \). So, the flux \( \Phi \) through the loop is given by:\[ \Phi = B \cdot A = \frac{\mu_0 i}{2a} \times \pi a^2 = \frac{\mu_0 \pi a i}{2} \]

Step 2: Formulate and Explain Faraday’s Law for Part (b)

According to Faraday's Law, the induced emf \( \mathcal{E} \) in the loop is \( \mathcal{E} = - \frac{d\Phi}{dt} \). Using the result from Step 1, \( \Phi = \frac{\mu_0 \pi a i}{2} \), calculate:\[ \mathcal{E} = - \frac{d}{dt} \left( \frac{\mu_0 \pi a i}{2} \right) = - \frac{\mu_0 \pi a}{2} \frac{di}{dt} \]By Ohm's Law, \( \mathcal{E} = iR \). Combine to form the differential equation:\[ iR = - \frac{\mu_0 \pi a}{2} \frac{di}{dt} \]Rearrange to:\[ \frac{di}{dt} = - \frac{2R}{\mu_0 \pi a} i \]

Step 3: Solve the Differential Equation for Part (c)

The differential equation derived is:\[ \frac{di}{dt} = - \frac{2R}{\mu_0 \pi a} i \]This equation is separable and can be solved by integrating both sides. Separate variables and integrate:\[ \int \frac{1}{i} di = - \int \frac{2R}{\mu_0 \pi a} dt \]This leads to:\[ \ln(i) = - \frac{2R}{\mu_0 \pi a} t + C \]Solving for \(i\), we find:\[ i(t) = i_0 e^{-\frac{2R}{\mu_0 \pi a} t} \] where \( i_0 \) is the current at \( t=0 \).

Step 4: Determine the Time for Part (d)

We need to find the time \( t \) when the current \( i(t) \) becomes \( \frac{i_0}{100} \). Using the expression from Step 3:\[ \frac{i_0}{100} = i_0 e^{-\frac{2R}{\mu_0 \pi a} t} \]Simplifying, we get:\[ e^{-\frac{2R}{\mu_0 \pi a} t} = \frac{1}{100} \]Taking the logarithm of both sides:\[ -\frac{2R}{\mu_0 \pi a} t = \ln\left(\frac{1}{100}\right) \]Solving for \( t \):\[ t = -\frac{\mu_0 \pi a}{2R} \ln\left(\frac{1}{100}\right) \]Substitute \( a = 0.5 \) m and \( R = 0.10 \) \( \Omega \), and calculate \( t \).

Step 5: Explanation for Part (e)

In practice, the effects of a self-induced emf due to changes in the current are often negligible because the committed changes in the magnetic field and hence the induced emf are small compared to primary external changes. Moreover, in many contexts, changes are rapid enough that their influence is minimal compared to other dominating factors like larger scale variations in external flux.

Key Concepts

Faraday's Law

Faraday's Law of electromagnetic induction is a fundamental principle that describes how electric currents are induced by changing magnetic fields. When a loop of wire is exposed to a changing magnetic field, a voltage, or electromotive force (emf), is generated around the wire. This phenomenon is called electromagnetic induction. The law is mathematically expressed as:

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

where \( \mathcal{E} \) is the induced emf and \( \frac{d\Phi}{dt} \) represents the rate of change of magnetic flux. In the given exercise, the loop experiences a decrease in external magnetic field, leading to a change in magnetic flux. It results in an induced emf that generates a current in the loop. The negative sign in Faraday's Law represents Lenz's Law, which states that the direction of induced emf opposes the change in flux that produced it. This is a key concept for understanding how electromagnetic induction works and how devices like generators and transformers function.

Magnetic Flux

Magnetic flux (\( \Phi \)) is a measure of the quantity of magnetism, taking into consideration the strength and the extent of a magnetic field. It is defined as the product of the magnetic field (\( B \)) and the area it penetrates (\( A \)). Mathematically, it is:

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

where \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface area. In the context of the exercise, we assumed a constant magnetic field throughout the loop for simplicity. By calculating the flux for varying currents, we capture how the loop's self-generated magnetic field contributes to self-induction. This self-induced flux occurs because the current creates its own magnetic field, influencing the loop even when the external field ceases.

Differential Equations

Differential equations are equations that involve the derivatives of a function. They are essential in describing the changes in physical systems, like the change in current in this exercise. For the loop, the relationship between the rate of change of current and the induced emf is expressed as a first-order differential equation:

• \( \frac{di}{dt} = - \frac{2R}{\mu_0 \pi a} i \)

This type of equation is known as a linear ordinary differential equation with constant coefficients. Solving it involves separation of variables and integration, yielding a solution that describes the decay of the current over time. This method allows us to understand how systems respond to changes, predicting their behavior based on initial conditions.

Ohm's Law

Ohm's Law is a basic principle relating voltage (\( V \)), current (\( I \)), and resistance (\( R \)) in an electrical circuit. It states:

• \( V = IR \)

In the exercise, Ohm's Law applies to the loop by equating the induced emf (\( \mathcal{E} \)) to the product of induced current and resistance. This gives us the connection between the changing magnetic flux and the current that flows due to it. By combining Ohm's Law with Faraday's Law, we derive the differential equation that describes how current varies as a result of self-inductive properties. Understanding this relationship helps to reconcile how generated currents affect the circuit and how they counterbalance the changes in magnetic fields.

Self-Inductance

Self-inductance is the property of a circuit whereby a change in the current flowing through it induces an emf within the circuit itself. It arises from the magnetic field generated by the current, affecting the entire loop. In the exercise, the self-inductance of the loop results in a current that continues even after the external magnetic field becomes constant. This is because the internal magnetic field generated by the current also changes the flux, inducing an emf that opposes the change. This behavior is captured by Faraday's Law and reinforces the loop's current to gradually decline rather than stop instantly. The concept of self-inductance is crucial for the understanding of inductors within circuits and is measured in Henrys (H), providing insights into the transient responses of AC circuits, power supply fluctuations, and more.