Problem 20 - University Physics with Modern Physics

Question

An electric field of magnitude \(200.0 \mathrm{~V} / \mathrm{m}\) is directed perpendicular to a circular planar surface with radius \(6.00 \mathrm{~cm}\). If the electric field increases at a rate of \(10.0 \mathrm{~V} /(\mathrm{m} \mathrm{s}),\) determine the magnitude and the direction of the magnetic field at a radial distance \(10.0 \mathrm{~cm}\) away from the center of the circular area.

Step-by-Step Solution

Answer

Answer: The magnitude of the induced magnetic field is \(7.96 \times 10^{-8} \mathrm{T}\) and the direction is upwards.

Step 1: Calculate the induced EMF

First, we need to calculate the induced electromotive force (EMF) in the circular planar surface due to the change in the electric field. According to Faraday's law of electromagnetic induction: \[\oint _C \mathbf{E} \cdot d\bold{ \ell} = -\frac {d\Phi _B}{dt}\] Where \(\mathbf{E}\) is the electric field, \(d\bold{ \ell}\) is the infinitesimal length element of a closed curve \(C\), and \(\frac {d\Phi _B}{dt}\) is the time derivative of the magnetic flux \(\Phi _B\). In this case, since the electric field is uniform and perpendicular to the circular area, the magnetic flux is: \[\Phi_B = E\pi r_e^2\] Where \(r_e = 6.00\mathrm{~cm} = 0.06\mathrm{~m}\) The time derivative of the magnetic flux is given by: \[\frac {d\Phi _B}{dt} = \frac {d(E\pi r_e^2)}{dt} = \pi r_e^2 \frac {dE}{dt}\] Then, we can substitute with the given values. \[\frac{d\Phi_B}{dt} =\pi (0.06\mathrm{~m})^2 (10.0 \mathrm{~V}/(\mathrm{m}\mathrm{s}))\]

Step 2: Calculate the induced current

Now, we need to find the induced current in the circular surface. To do that, we can rewrite Faraday's law as follows: \[I \oint _C \mathbf{B} \cdot d\bold{ \ell} = -\frac {d\Phi _B}{dt}\] Where \(I\) is the induced current and \(\mathbf{B}\) is the magnetic field. Since the direction of the induced current is tangential to the surface and perpendicular to the magnetic field, the expression for the induced current becomes: \[I = -\frac {1}{2\pi r_c}\frac {d\Phi _B}{dt}\] Where \(r_c = 10.0\mathrm{~cm} = 0.10\mathrm{~m}\) is the radial distance away from the center of the circular area.

Step 3: Calculate the induced magnetic field

Now, we will use the Biot-Savart law to find the magnetic field induced by the induced current at radial distance \(r_c\). The Biot-Savart law is given as follows: \[\mathbf{B} = \frac{\mu_0}{4\pi}\int \frac{I{}d\mathbf{l} \times \mathbf{r'}}{r'^3}\] For this circular surface, the induced magnetic field is: \[\mathbf{B} = \frac{\mu_0 I}{2r_c}\] Where \(\mu_0\) is the permeability of free space. Now, we can substitute the induced current and the given values: \[\mathbf{B} = \frac{(4\pi \times 10^{-7}\mathrm{Tm/A})}{2\cdot 0.10\mathrm{~m}}\cdot \left(-\frac {1}{2\pi 0.10\mathrm{~m}}\right)\frac {d\Phi _B}{dt}\]

Step 4: Calculate the magnitude and direction of the induced magnetic field

Now, we can find the magnitude of the induced magnetic field, with direction given by the right-hand rule. Using the given values and the result from the previous steps: \[\mathbf{B} = 7.96 \times 10^{-8} \mathrm{T}\] The induced magnetic field is upwards if we take into account the right-hand rule and the direction of the electric field. Final answer, Magnitude: \(7.96 \times 10^{-8} \mathrm{T}\) Direction: Upwards

Key Concepts

Faraday's Law

While electromagnetic induction can seem daunting, Faraday's Law provides a foundation to understand it simply. Faraday's Law of electromagnetic induction states that a change in the magnetic environment of a coil of wire will cause a voltage (or emf) to be induced in the coil. This means whenever the magnetic flux through a closed surface is varied, an electromotive force (EMF) is created. In a mathematical form:\[ \oint_C \mathbf{E} \cdot d\bold{ \ell} = -\frac {d\Phi _B}{dt} \]where \(\mathbf{E}\) is the electric field, \(d\bold{\ell}\) is a small segment of the closed loop \(C\), and \(\frac{d\Phi_B}{dt}\) represents the change in magnetic flux over time. In our exercise, the changing electric field given by \(10.0 \mathrm{~V}/(\mathrm{m}\mathrm{s})\) directly affects the magnetic flux, leading to an induced EMF. This shows how electric and magnetic fields are interconnected in dynamic environments.

• The direction of induced EMF (and hence current) opposes the change in magnetic field - often called Lenz's Law.
• This foundational principle is used in various applications, like electric generators and transformers.

Biot-Savart Law

Understanding the Biot-Savart Law helps us calculate the magnetic field resulting from an electric current. The law essentially relates the magnetic field produced around a wire to the current flowing through the wire. It provides a cumulative technique to sum up contributions to the total magnetic field from all parts of the wire:\[ \mathbf{B} = \frac{\mu_0}{4\pi}\int \frac{I d\mathbf{l} \times \mathbf{r'}}{r'^3} \]In this exercise, we're applying a simplified version of the Biot-Savart Law for a circular loop to calculate the magnetic field at a point some distance away. Since the current induced by changing electric fields creates a loop (like in our case), this law is useful in solving for the magnetic field strength. This provides insight into:

• How current loops generate fields, forming the basis for electromagnetism.
• The importance of parameters like current direction and distance from the wire.

Magnetic Field Calculation

To calculate the magnetic field's magnitude and direction, several physics concepts are intertwined. In our case, we first derive the induced current using Faraday's law and then apply the Biot-Savart law. The formula used reflects direct contribution from the induced current to the field:\[ \mathbf{B} = \frac{\mu_0 I}{2 r_c} \]where \(\mu_0\) is the permeability of free space, and \(I\) is the induced current over distance \(r_c\). By substituting our induced current in the expression, we find the magnitude of the field. The right-hand rule can then be used to determine direction.
Key points include:

• Both the current and radius significantly impact the field's strength.
• Understanding directional properties of magnetic fields is essential.

Induced EMF

Induced EMF is the driving voltage generated within a closed circuit due to changing magnetic fields, as stated by Faraday’s Law. This phenomenon allows conversion of magnetic energy into electrical energy. By altering the external magnetic field, an internal EMF starts to induce within the conductor.In our exercise, induced EMF arises from the time-varying electric field over the circular area:\[ \frac{d\Phi_B}{dt} = \pi r_e^2 \frac {dE}{dt} \]This relationship shows how rate of change in electric field can transform into a usable voltage, valid within devices like transformers.
Some takeaway points:

• Induced EMF is predicted by the rate of change in magnetic flux.
• Explains how mechanical motion can transform into electric energy.