The Wireless Charging Coil · AI Generator #GEN-09EE89B2

Physics · electromagnetism

Problem

A smartphone uses wireless charging technology based on electromagnetic induction. The charging pad contains a flat circular coil of radius

R = 4.0\;\text{cm}

with

N_1 = 100

turns carrying an alternating current

I(t) = I_0 \sin(\omega t)

where

I_0 = 2.0\;\text{A}

and

\omega = 2\pi \times 85\;\text{kHz}

. The phone contains a smaller receiving coil of radius

r = 2.0\;\text{cm}

with

N_2 = 50

turns, positioned parallel to and directly above the charging coil at a distance

d = 5.0\;\text{mm}

. For two coaxial circular coils separated by distance

d

, where

d

is small compared to the coil radii, the mutual inductance is approximately:

M \approx \frac{\mu_0 N_1 N_2 \pi r^2 R^2}{2(R^2 + d^2)^{3/2}}

where

r

is the radius of the smaller coil and

R

is the radius of the larger coil. Find the peak voltage induced in the receiving coil. Use

\mu_0 = 4\pi \times 10^{-7}\;\text{H/m}