MacBook – Do all magsafe power ports have the same form factor

Messing with a line voltage components like transformers is a huge fire hazard.

When these cords fail due to mechanical abrasion, it's common for other parts to melt or fray and I wouldn't trust myself to know if the repaired unit would be safe. I would worry the heat of soldering would do more damage to the cord close to the solder point.

Have you already called Apple and had them check if that adapter can be exchanged?

The MagSafe power adapter, like nearly all power adapters for modern electronic equipment, is a switching (switched-mode) power supply.

Switching power supplies achieve their light weight and small size by using smaller, lighter transformers for voltage conversion, which require high-frequency AC input to operate. This high-frequency input is produced by a separate circuit which rapidly switches the low-frequency input on and off (hence “switching”), and in fact does so controlled by feedback from the DC output side in order to achieve a tightly-regulated DC output. This regulation itself is also something that varies over time as the input and the output load change.

So, there's lots of varying electrical currents, and therefore varying electric and magnetic fields, inside a switching power supply. These fields exert mechanical forces on the components within the power supply, which may cause them to vibrate and therefore produce sound!

In a well-designed power supply, the basic frequencies are chosen to be above the range of human hearing (and the components most subject to vibration may be held down especially firmly). However, for various reasons (waves hands, I don't actually know the details) you may get effects at fractions of those frequencies which you can actually hear.

In your particular case, if the power supply is running overloaded (or at the upper limit of its capability), then that might put some subsystem oscillating at the upper end of audibility, producing the sort of noise you're hearing.

You can often hear noise from cheap power supplies (e.g. USB chargers) when they are unloaded (nothing connected to the output), and hear it shift out of audible range when you connect something.

(Disclaimer: I Am Not An Electrical Engineer; this is just bits and pieces I've picked up. These folks would have the whole story; e.g. this answer which is a bit short on detail. But, I hope I've at least somewhat demystified why apparently solid lumps of electronics can make noise. (If anyone sees a factual error or can explain better, please feel free to edit this answer!))