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Below are photos of two scope probes with different ground clip lengths:

I have read that the shorter ground is used to minimize the inductance of the probe ground lead.

But what does that help for? What happens when the inductance of the ground lead is low? What kind of interference it prevents?

It doesn't prevent interference. It prevents ground lead impedance.

Simply imagine an inductor in series with your ground connection: that acts as a low-pass filter. So, high-speed currents can't be grounded, and for these, your instrument seems to float.

I was invited to assist in debugging a switching regulator IC; problem was "two kinds of oscillation".

I asked what were the frequency of oscillation, and answer was 80 MHz.

I asked "how long is the scope ground lead" with answer being "The usual 6 or 8 inches".

I explained "The resonant frequency, of 200 nH (8") gnd-lead scope probe with 15 pF input capacity, is about 90 MHz."

Turns out the silicon designer had cranked out LDOs in his prior IC work, and had never needed to learn fast transient probing methods. Here he got to learn about scope probe ringing.

The other form of oscillation/noise/weird-behavior involved jitter in the timing of entering and exiting discontinuous modes. That involved very very slow decays of the regulated voltage and timing errors caused by thermal noise.

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What is the resonant frequency of the coiled springy Ground structure, pushed onto the Ground ferrule? Ignore the possibility of poor contact, where the numerous turns scale up the inductance. In other words, assume the path length is 1 cm of Center plus 1 cm of GROUND return, or 2 cm total or about 20 nH total. This is a good assumption, because the formula for inductance is Constant * Length * (1 + log(length/wiresize)), causing the computed inductance to be a mostly linear function of length.

What is the resonant frequency of 20nH and 15pF?
I use

(F_MHZ)^2 == 25,330 / (L_uH * C_pf)

where 1 uH and 1 pF => F_MHz = sqrt(25,330) = 160 MHz

We have 0.02 uH and 15 pF, with product of 0.3.

Divide that into 25,330, with quotient of 75,000.

The squareroot is about 280 MHz.

How about improving that ringing? Can we dampen? Yes.
Add an external discrete resistor at the probe tip.
Value?
Go for Q=1, thus Xl = Xc = R.
Xc of 15 pF at 280 MHz, given 1 pF at 1 GHz is -j160 ohms, is 160/15 * 1,000 MHz/280 MHz or approx. 30 ohms.

What does this do to the probe behavior at high frequency? The Trise will be approx. 15 pF * 33 ohms, or about 0.45 nanoSec or 450 picoSec. Fast enough?
Just grab a 33 ohm discrete resistor and use needle nose pliers to crimp that resistor lead around the center pin of the probe tip.

And there should be no ringing at the 280 MHz Fring.

There are three effects two consider here.

1. Transformer Action (H-Field): Any loop within a changing magnetic field gets a voltage induced into it. This is the idea behind transformers. A long can see more flux so is more susceptible to magnetic pickup.




2. Capacitive Effects (E-Field): Any two conductors separated by an insulator for a capacitor. Since $ C = dfrac{epsilon cdot A}{d} $ having a shorter wire reduces the are of one the plates and hence the capacitance reducing E-Field sensitivity.




3. Ground lead inductance: As pointed out by Marcus inductance in the ground lead increases the impedance to high frequency signals and a longer wire has more inductance. You can also reduce the inductance by wrapping the ground tight to the probe but it is less good than you have shown in your second picture.

Which of these dominates depends on the circuit you are testing. I regularly connect the the ground lead of my probe to the tip of the probe. This should should see nothing as you are measuring the 0V of the scope. However, It will show you where there are significant magnetic fields in your circuit.