The x1 and x10 switch

They call it x10, but actually its divide by 10 as the probe has a series resistor in it in x10 mode.  The x1 mode switch simply bypasses this series resistor.  If you want to know all the details take at look at this excellent video.

Probes are normally used with the probe switch set to x10 as this gives the maximum specified bandwidth.  If you look at a probe datasheet it will specify the maximum bandwidth for each switch setting, e.g.:

x1: DC~8MHz

x10: DC~350MHz

The max rise time will also be slower for the x1 switch setting.

x1 Setting

Has a much higher input capacitance than x10, so it loads your signal much more at high frequencies.  The only resistance is that of the Coax cable, but bear in mind that oscilloscopes use a lossy coax cable which has a significant resistance.  This is because this cable is working as a transmission line and scopes are optimized around this and x10 probe use.   The advantage of using x1 mode is that you are not attenuating your signal by 10x and the option to switch to this mode is provided for situations where that might be important.

x10 Setting (Default)

Gives you the full bandwidth of the probe and is what the scope is optimized around so you can measure fast signals.

Some probes have no switch and are hardwired to x10, which gives a slight edge over the switchable type as the switch compromises the signal slightly.

Are expensive probes worth the money?

Scope probes use a lossy coax cable that requires special manufacturing to make it work as a perfect transmission line.  The higher the bandwidth of the probe the tighter the tolerances and the cleverer the manufacturing process required to make the cable (and other probe parts) so the higher the cost.

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