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When you take the cable out of your laptop and plug it in to the router, the router requests
an IP address using DHCP (dynamic host configuration protocol)
The IP address you got using your laptop is tied to the MAC address of your network adapter.
When the cable gets plugged in to a different network adapter (your router internet port and it's MAC address) it gets a different
No. DHCP on the router only hands out IP addresses for the local LAN.
When you set up the router you specified your type of internet connection. In the case of cable companies, they use DHCP whereas a DSL connection utilizes PPPoE.
If you look at the Properties for the network adapter in your laptop you will see that it's set to automatically obtain an IP address (that's DHCP)
If you were to connect your laptop to the router and go into the router configuration pages and clone the MAC address of the laptop into the router the IP address would stay the same when you switch the cables.
In that situation, no matter if the laptop were directly connect to the modem or connected to the router first. The cable modem would see the same MAC address.
There is no public IP address assigned to the modem. It passes the public IP address to the device attached to it.
Any time a network adapter is connected to a network it makes a DHCP request for an IP address. In the case of a cable customer the request
is sent to the providers router which issues the IP address. IP addresses are leased. Meaning they only last for a certain amount of time. Typically 3 to 7 days for cable companies though they can be set to any length.
They can also be renewed which can result in the issuance of a different IP address.