I would recommend using a wire gauge that is appropriate for the frequency of your LCR resonant circuit. For example, if you are working with a high frequency circuit, a thinner wire gauge would be more suitable, while a lower frequency circuit may require a thicker wire gauge.
To wind the coil within 5% variation, I suggest using a coil winding machine or a hand-held coil winder to ensure consistent and accurate winding. It is important to maintain the same tension and spacing between each turn of the wire to achieve a uniform coil.
In terms of calculating the number of turns needed for a 10mH inductor, you can use the formula L = (μN^2A)/l, where L is the inductance in henries, μ is the permeability of the core material, N is the number of turns, A is the cross-sectional area of the core, and l is the length of the core.
To achieve a more precise result, you can also use an inductance calculator or refer to a manufacturer's datasheet for the core material you are using. Alternatively, you can also measure the inductance of the coil using an LCR meter to ensure it is within the desired range.
I hope this helps guide you in creating your 10mH inductor coil for your LCR resonant circuit. If you have any further questions or need clarification, please do not hesitate to reach out. Best of luck in your project!