How does space charge limit the flow of electrons from a heated cathode?

In an electron tube, space charge limits the flow of electrons through the formation of a negative electron cloud near the cathode. Because emitted electrons do not travel instantaneously to the anode, they accumulate in the region between the electrodes.12

This cloud of negative charge exerts a repulsive force on newly emitted, low-energy electrons, causing them to return to the cathode rather than continuing toward the anode. This phenomenon is known as space-charge-limited operation.23

The flow of electrons can be increased by applying a positive voltage to the anode, which attracts electrons from the cloud. As the voltage rises, more electrons are pulled toward the anode until the device reaches saturation, where all emitted electrons are collected. The current density in this space-charge-limited region is mathematically described by the Langmuir-Child law.2