What condition is necessary for the flow of heat? What analogous condition is necessary for the flow of charge?
A difference in temperature is needed for heat to flow. A difference in electrical potential is needed for charge to flow. The flow of charge persists as long as there is potential difference ; without a potential difference, or once both ends reach an equilibrium, flow ceases.

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What condition is necessary for the sustained flow of water in a pipe? What analogous condition is necessary for the sustained flow of charge in a wire?
A continuous pressure difference, often provided by a pump, is needed for water to flow. A continuous potential difference, often provided by a battery, is needed for charge to flow. Once a common potential is reached, or difference ceases, flow ceases. Charge flow is actually particle* flow. Negative charge flow= electrons, negative ions. Positive flow = protons or positive ions are flowing.
Electrons are free to move through the metal, whereas protons are fixed in place.In metal wires, 1+ e- from each metal atom are free to move throughout the atomic lattice as conduction electrons white protons are bound inside nuclei. In conducting fluids like car batteries, positive ions are part of the charge flow.
measurement unit for rate of electrical flow. One ampere = 1 coulomb of charge per second. *1 coulomb = 6.25 X 10^18 e-
batteries and automobile alternators (electric generator). These do work to pull negative charges away from positive ones. By chemical disintegration or electromagnetic induction. The different values of energy per charge create potential difference.Van de Graaff is not practical! (single brief surge of voltage)
12 joulesex. common automobile battery provides 12V electrical pressure to circuit. 12 J energy are supplied to each coulomb of charge made to flow in the circuit.
Does electric charge flow across a circuit or through a circuit? Does voltage flow across a circuit or is it impressed across a circuit?
Charge flows through and voltage is impressed across a circuit.*Water flows through and pressure changes across the the ends.

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Will water flow more easily through a wide pipe or a narrow pipe? Will current flow more easily through a thick wire or a thin wire?
Heated metal wires increase resistance because atoms at higher temperatures jostle into the way of moving electrons.
If the voltage impressed across a circuit is held constant while the resistance doubles, what change occurs in the current?
It is cut in half.current = voltage/resistanceamperes = volts/ohms"the greater the voltage, the greater the current)(the greater the resistance, the smaller the current)
If the resistance of a circuit remains constant while the voltage across the circuit decreases to half its former value, what change occurs in the current?
It is cut in half."the greater the voltage, the greater the current"∴ "the smaller the voltage, the smaller the current"