High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Physics Textbook (Master) > Magnetic Induction

Faraday's Law of Induction - with calculus

Faraday's Law of InductionPrevious Section
Lenz's LawNext Section
Popular Courses
Find My Course
0:00 / 0:00

Faraday's Law


Often, we need to use calculus to fully explore problems with Faraday's Law.
  • Faraday's Law can be re-written with calculus by replacing the rate of change with a derivative:

ε=∣dΦmdt∣=∣ddt(BAcos⁡θ)∣\boxed{\varepsilon =\bigg|\cfrac{d \Phi_m}{dt}\bigg|=\bigg|\cfrac{d}{dt}(BA \cos \theta)\bigg|}

  • With this version of Faraday's Law, we can use the product rule to see what happens if more than one variable is changing.
  • For example, if the magnetic field strength B(t)B(t) and area A(t)A(t)are both functions of time, Faraday's Law would be written as follows:
ε=∣ddt(B(t)A(t)cos⁡θ)∣ε=∣(B(t)dAdt+A(t)dBdt)cos⁡θ∣\begin{aligned} \varepsilon &=\bigg|\cfrac{d}{dt}(B(t)A(t) \cos \theta)\bigg| \\ \varepsilon &=\bigg|\bigg(B(t)\cfrac{dA}{dt}+A(t)\cfrac{dB}{dt}\bigg)\cos \theta\bigg| \\ \end{aligned}
  • If the flux needs to be determined with an integral, the most general form of Faraday's Law can be written as follows:
ε=ddt(∫SB⃗⋅dA⃗)\boxed{\varepsilon=\cfrac{d}{dt}(\int_S \vec B \cdot d \vec A )}

Practice: Non-Uniform, Time-Dependent Magnetic Field


A square loop of wire with sides of length aa lies in the positive quadrant of the xy-plane with one corner at the origin of the coordinate system. There is a magnetic field present given by the following expression:
B⃗(x,y)=β(xcos⁡ωti^+ysin⁡ωtk^)B⃗(x,y)=(βxcos⁡ωt,0,βysin⁡ωt)\vec B(x,y)=\beta(x\cos\omega t \hat i+y\sin\omega t \hat k)\\ \vec B(x,y)=(\beta x\cos\omega t ,0,\beta y\sin\omega t)
a) What are the units of β\beta?
b) Determine the induced emf as a function of time.
Part a)

Practice: Two Rings of Current


Two circles of wire are located on the same plane and have the same center point, as shown below.

The outer loop has a radius of aa, and the radius of the inner loop is a8\frac{a}8. The outer loop has a linearly increasing current of I(t)=βtI(t)=\beta t.

a) What is the magnitude of the magnetic field created at the center point by the outer current loop?
b) Assume that the magnetic field at the center point from part (a) is the only magnetic field passing through the inner loop. What is the induced current in the inner loop if it has a resistance of RR?
Part a)

Practice: Expanding Triangle


Two conducting wires are located in a uniform magnetic field BB that points out of the page. The two wires are at an angle of 60 degrees. A vertical bar with resistanceRR with constant speed vv moves along the axis in the positive x-direction as shown below.

What is the induced current in the bar as a function of xx?