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Many dishwashers ( but not all ) use a BI-metal strip that heats up once power is applied to it. The BI-metal strip bends and allows the soap door spring to open up the dispenser. This bi-metal strip is not given full 120 volts power, the timer "shunts" the power through either the heating element ( most common ) or the motor and this gives the bi-metal strip lower than full power to work and open. Most often the bi-metal strip has lost it's ability to bend properly and needs to be replaced. Older GE dishwashers used an mechanical arm driven by the timer to open the dispenser = no electrical bi-metals. Newer ( plastic liner ) Maytag dishwashers and many newer Whirlpool and Frigidaire dishwashers use a wax motor to open the soap dispenser, this wax motor does receive 120 volts to work. The plunger inside the wax motor slowly moves and pushes the plunger and door open. This is a what a wax motor looks like.
My GE ( HP, Moff, McClary ) dishwasher soap door will not stay closed: ( Round style dispenser ) You should remove power and check the spring and plastic tab under the timer, often the plastic tab breaks and the soap door will not stay closed. You can try to install a small screw where the tab used to be to re-hook the spring. If that is ok, you will have to check the rest of the dispenser arms, door cover and handle. The spring on the round dispenser should look like this. #1 hooks on a groove cut into the dispenser arm, #2 hooks onto a tab built onto the inner door panel.