Choosing the right transistor for a switch
You'd be better to use a MOSFET for this kind of application. 

R1 keeps the Arduino output current low during switching (the MOSFET gate looks a bit like a short during the brief switching interval), and R2 makes sure that if the drive gets 
disconnected the MOSFET switches 'off' and does not hang around half-off and half-on, burning itself up. 







simulate this circuit &ndash; Schematic created using CircuitLab

MOSFETs have almost infinite current gain, so the \$h_{FE}\$ equation does not apply, but you do have be sure that there is enough drive voltage to turn the MOSFET well on and off. 

In this case, the maximum on resistance of the MOSFET with 4.5V drive is 11.9m\$\Omega\$, so the power dissipation will be only 2mW, and it will run dead cold.