There are many components both inside and outside of a cell phone that allow it to work properly. Inside, there is a battery that provides electricity, and a SIM card that allows it to connect to the cell phone tower. On top of the cell phone, a small antenna must be able to transmit signals to the cell phone tower. By disrupting any or all of these pathways, a cell phone can be disabled.
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Remove the battery. This is the simplest method, because without the battery any cellphone is nothing more than a brick of plastic and silicon. .
Remove the SIM card. The SIM card is the "key" that tells the cell phone tower that the cell phone is linked to a cell phone plan that has been paid for. Without the card, no calls can be made besides 911 calls. Note that there are ways to get around this: many convenience stores sell prepaid SIM cards that come with minutes and/or text messages.
Add a digital lock to the phone. Not all phones have the feature, but many do. Some phone allow you to lock your SIM card and your phone separately, so that even if someone has a prepaid SIM card they still need to password to unlock the phone. This is perhaps the best way to disable and secure a phone.
Interfere with the radio frequency that the cell phone uses. Cell phones operate on very specific radio frequencies, so a radio jammer can very precisely disrupt cell phone signals without affecting other appliances, such as household cordless phones.