Question 18.5: Determine the B field produced by an electromagnet To determ......
Determine the \vec{\pmb{B}} field produced by an electromagnet
To determine the \vec{B} field produced by an electromagnet, you use a 30-turn circular coil of radius 0.10 m (30-Ω resistance) thats rests between the poles of the magnet and is connected to an ammeter. When the electromagnet is switched off, the \vec{B} field decreases to zero in 1.5 s. During this 1.5 s the ammeter measures a constant current of 180 mA. How can you use this information to determine the initial \vec{B} field produced by the electromagnet?
Sketch and translate
■ Create a labeled sketch of the process described in the problem. Show the initial and final situations to indicate the change in magnetic flux.
■ Determine which physical quantity is changing (\vec{B}, A, \text { or } \theta) , thus causing the magnetic flux to change.
The changing quantity (from time 0.0 s to 1.5 s) is the magnitude of the \vec{B} field produced by the electromagnet. Due to this change, the flux through the coil’s area changes; thus there is an induced current in the coil. We can use the law of electromagnetic induction to find the magnitude of the initial \vec{B} field produced by the electromagnet.
Simplify and diagram
■ Decide what assumptions you are making: Does the flux change at a constant rate? Is the magnetic field uniform?
■ If useful, draw a graph of the flux and the corresponding induced emf-versusclock reading.
■ If needed, use Lenz’s law to determine the direction of the induced current.
Assume that
■ The current in the electromagnet changes at a constant rate, thus the flux through the coil does also.
■ The \vec{B} field in the vicinity of the coil is uniform.
■ The \vec{B} field is perpendicular to the coil’s surface.
Represent mathematically
■ Apply Faraday’s law and indicate the quantity (\vec{B}, A, \text { or } \cos \theta) that causes a changing magnetic flux.
■ If needed, use Ohm’s law and Kirchhoff’s loop rule to determine the induced current.
\varepsilon_{\text {in }}=N\left|\frac{\Phi_{\mathrm{f}}-\Phi_{\mathrm{i}}}{t_{\mathrm{f}}-t_{\mathrm{i}}}\right|=N\left|\frac{0-B_{\mathrm{i}} A \cos \theta}{\Delta t}\right|
The number of turns N and the angle θ remain constant. The magnitude of the magnetic field changes.
Substitute \varepsilon_{\text {in }}=I R \text { and } A=\pi r^2 \text { and solve for } B_{\mathrm{i}} \text { : }
B_{\mathrm{i}}=\frac{I R \Delta t}{N \pi r^2 \cos \theta}
Solve and evaluate
■ Use the mathematical representation to solve for the unknown quantity.
■ Evaluate the results—units, magnitude, and limiting cases.
The plane of the coil is perpendicular to the magnetic field lines, so θ = 0 and cos 0° = 1. Inserting the appropriate quantities:
B_{\mathrm{i}}=\frac{I R \Delta t}{N \pi r^2}=\frac{0.18 \mathrm{~A} \times 30 \Omega \times 1.5 \mathrm{~s}}{30 \times \pi \times(0.1 \mathrm{~m})^2}=8.6 \mathrm{~T}
This is a very strong \vec{B} field but possible with modern superconducting electromagnets. Let’s check the units:
\frac{A. \Omega.s}{m^2}=\frac{V.s}{m^2}=\frac{J.s}{C.m^2}=\frac{N.m.s}{C.m^2}=\frac{N}{C.(m/s)}=TThe units match. As a limiting case, a coil with fewer turns would require a larger \vec{B} field to induce the same current. Also, if the resistance of the circuit is larger, the same \vec{B} field change induces a smaller current.
Try it yourself: Determine the current in the loop if the plane of the loop is parallel to the magnetic field. Everything else is the same.
Answer: Zero.