Given:
- Magnetic field strength (B) = 1 T

To find:
Cyclotron frequency of an electron gyrating in the magnetic field

Solution:
The cyclotron frequency (ω) is the frequency at which a charged particle completes one revolution or gyrates in a magnetic field.

The formula to calculate the cyclotron frequency is given by:
ω = qB/m
where q is the charge of the particle and m is its mass.

In this case, we are considering an electron, so the charge of the electron (q) is equal to -1.6 x 10^-19 C (coulombs) and the mass of the electron (m) is equal to 9.11 x 10^-31 kg.

Plugging in the values into the formula, we get:
ω = (-1.6 x 10^-19 C) x (1 T) / (9.11 x 10^-31 kg)

Simplifying the expression:
ω = -1.76 x 10^11 C.kg^(-1).T^(-1)

Converting the units:
1 C = 1/(s.A) (coulombs)
1 kg = 10^3 g
1 T = 10^4 G (teslas to gauss)

ω = -1.76 x 10^11 (1/(s.A)) x (10^3 g)^(-1) x (10^4 G)^(-1)

Simplifying the expression:
ω = -1.76 x 10^11 (s.A)^(-1) x g^(-1) x G^(-1)

Further simplification:
1 A = 1 C/s (ampere)
1 G = 10^(-4) T (gauss to teslas)

ω = -1.76 x 10^11 (C.s)^(-1) x g^(-1) x (10^(-4) T)^(-1)

Simplifying the expression:
ω = -1.76 x 10^11 (C.s)^(-1) x g^(-1) x 10^4 T^(-1)

Converting back to base units:
1 s = 10^9 ns (seconds to nanoseconds)
1 C = 10^9 nC (coulombs to nanocoulombs)

ω = -1.76 x 10^11 (10^9 ns)^(-1) x g^(-1) x 10^4 (10^9 nC)^(-1)

Simplifying the expression:
ω = -1.76 x 10^11 (ns)^(-1) x g^(-1) x 10^4 (nC)^(-1)

Finally, converting to the desired unit of frequency (GHz):
1 GHz = 10^9 Hz (gigahertz to hertz)

ω = -1.76 x 10^11 Hz x g^(-1) x 10^4 Hz^(-1)

Simplifying the expression:
ω = -1.76 x 10^11 x g^(-1) x 10^4

Calculating the value:
ω ≈ -1.76 x 10^11 x 10^4
ω ≈ -1.76 x 10^15 Hz