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How can armor, including shields, made with materials available in medieval times, be made to withstand magical fire and lightning attacks?
The fire attacks are equivalent to that from a propane flamethrower, and have a range of around 160-200 feet. The lightning attacks have roughly the power of a small Tesla coil, but can be aimed by the caster. Lightning attacks will still divert to a suitably conductive object.
A metal shield tower shield, insulated near the arm and head areas with wool, and trailing a short length of wire on the ground, would improve your chances against fire and electrical attacks. Your gauntlets would either need to be lined with rubber, or the grip of the shield lined with rubber, to insulate against the electricity. Note that this would only help with a directed attack and not an area-of-effect attack. You're cooked with an AoE attack. Best, then, to put an arrow in the magician's eye first.
If you know the magician has electrical spells and not fire, then a wooden shield would be much more valuable.
Finally, it's always worth noting that the attack volume or duration will always have the ability to overcome non-magical armor. throw enough fire or electricity at our intrepid plate-mail paladin, or throw it for a long enough period, and it won't matter how the shield is designed. It'll either become hot enough to ignite the wool and melt the rubber, or so over charged that it might become magnetic and stick to your cuirass, binding your arm along with it.
As long as your armor is metal, touching the ground, and there is a continuous conductive path, any metal armor will protect against lightning and electrical attacks until the current is sufficient to melt/vaporize the metal. It was not uncommon for "gaps" to be covered or linked to the main body of armor with chain (either mail or simple links).
Medieval armies had to deal with "greek fire", which can be thought of as a primitive napalm. Fire attacks such as you describe were commonplace on medieval battlegrounds, as were defenses against them.
There is no defense against such attacks other than "magic". If you try to take this by science, you'll fail.
The reason is simple:
a) Lightning doesn't work in such a way as it goes one particular way as pointed by the wizard, and damages the target. Lightning is electricity, so there must be a closed circuit. Current is everywhere within that circuit. Fullstop.
Lightning fired from a wizard's fingertip would damage the wizard as much (or more!) as the target. The current would go through the ionized air, then wander over the Faraday cage surrounding the target (metal armour) without causing significant damage, and close the circuit via the ground, thereby frying the wizard.
b) Magical fire (burning hands, dragon breath?) will hard boil anyone inside a piece of armor whether the armor stops the fire or not. Plus, it will render air unbreathable (both due to lack of oxygen and temperature). Metal has high thermal conductivity, but even with thick layers of insulation (try and move with one foot thick of bandages wrapped around your thighs), the temperature inside would rise beyond what's tolerable for a human anyway. Even with modern technology and with appropriate breathing equipment, standing in a definitively non-magical fire is (although tolerable for a while) already not a big pleasure. Add to that the fact that usually magical fire is thought to burn "extra hot" so the metal would melt anyway.
The same as with lightning applies for magical fire in respect of the wizard, too. Again, it works both ways. Fire shot from your fingertips will cause burns at your hands, and will possibly (likely, even) cauterize your lungs if you breathe in the hot air.
Unless, well, magic.
So the answer to appropriate armor is, again... magic. Some blessing, some special, rare magical metal mined by the dwarves of ere, metal that fell from the sky, whatever.
