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In these 3 circuits, I try to illustrate the use of a pull-down resistor. The same principle applies to a pull-up resistor, too. The middle one is broken, and the others work.
1) Push button with pull-down resistor.
When the button is pressed, 5V is applied to the MOSFET, turning it ON. When the button is released, the resistor ties (or connects) the MOSFET to 0V, turning it OFF. This is the basic action of a pull-down resistor.
2) Push button without pull-down resistor.
In this circuit, the pull-down resistor is removed. As you can see, the MOSFET doesn't work as we wanted. The reason for this is that the gate is said to be "floating" when the button is not pressed. Floating means that the voltage on that line is undefined (basically). The gate is disconnected (which does NOT mean 0V!). There is no resistor to attach the gate to anything when the button isn't pressed. When this happens, the MOSFET doesn't know whether to be ON or OFF. This can cause lots of problems in logic circuits where it is important to keep the voltage level at 1 or 0, not floating or unknown.
3) Logic source without pull-down resistor.
The push button has been replaced with a logic source. The circuit works correctly without a resistor to pull down the voltage when the logic source is OFF. Why? Because when the logic source is OFF, it is the same as 0V, in contrast to the button, which is high impedance (or disconnected) when it is released. So, there is no "floating" state, and the MOSFET turns ON and OFF correctly.
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