Arectangular loop of area a is placed in a region where the magnetic field is perpendicular to the plane of the loop. the magnitude of the field is allowed to vary in time according to b = bmax e-t/τ , where bmax and τ are constants. the field has the constant value bmax for t < 0. find the emf induced in the loop as a function of time. (use the following as necessary: a, bmax, t, and τ.) g

Induced emf,

Explanation:

The varying magnetic field with time t is given by according to equation as :

Where

are constant

Let is the emf induced in the loop as a function of time. We know that the rate of change of magnetic flux is equal to the induced emf as:

So, the induced emf in the loop as a function of time is . Hence, this is the required solution.

Explanation:

About 130 million years ago, two stars measuring 12 miles across but containing more mass than held in our Sun were locked together. They were spiralling together, in a relatively rare event that lasted only a few seconds and likely ended up with the creation of a black hole. But the event sent out a signal, in the form of gravitational waves and light. It also made a sound. Not a bang, but a kind of whizz-bang chirpy pop.

At the exact time this happened, dinosaurs roamed on planet Earth. When the signals from the merging neutron stars were about half way on their journey towards Earth, an asteroid struck our planet, killing off 75 per cent of all species.

Over the last five per cent of the gravitational waves’ journey, humans evolved, learnt how to use tools, developed civilisations and started looking to the skies with wonder.

Just 100 years ago, one human predicted the impact such colossal events would have on spacetime. Now we can detect them.

On the August 17, the Ligo team detected the ripples in spacetime given off by one of the most dramatic events that occur in our Universe; the merging of these two neutron stars.

When Ligo’s twin detectors, in Louisiana and Washington state, first saw the tremors in spacetime, they alerted astronomers across the globe. Within hours, 70 telescopes, both in space and on Earth, were pointing towards the galaxy 130 million light years away.

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Two seconds after Ligo watched this happen, the Fermi space telescope spotted a burst of gamma rays – and putting the two signals together, the chance of a coincidence was tiny. Within an hour, a third detector called Virgo, near Pisa in Italy, had confirmed the signal, and helped narrow down the section of the sky the signal had come from.

Scientists could tell this wasn't a pair of black holes, since the masses of the two swirling objects were between 1 and 1.6 times the mass of our Sun.

"It immediately appeared to us the source was likely to be neutron stars, the other coveted source we were hoping to see—and promising the world we would see," says David Shoemaker, spokesperson for the Ligo Scientific Collaboration and senior research scientist in MIT's Kavli Institute for Astrophysics and Space Research.

Another big difference between the black hole merger events and this one is the time it takes. Black hole mergers are much faster, with gravitational waves only lasting a fraction of a second. In this case, the ripples could be seen for 100 seconds.

The first gravitational waves were detected by Ligo as the neutron stars swirled around each other, coming closer and closer. From passing about 30 times each second, the pair reached a point around 100 seconds later when they were swirling 2,000 times each second.

After that, there was a short pause when nothing was detected by Ligo. Then, the stars violently collided, most likely creating a black hole; making this, probably, the first time we have witnessed a black hole being made.

The violent burst gave off not only gravitational waves, but also a short burst of gamma rays, and light in a huge range of the electromagnetic spectrum. Short gamma ray bursts have been seen before, and it was thought they were created by neutron star mergers. But this is the first time it’s been confirmed.

“We detected the gamma ray burst just because it was so close and the immediate hint about its small distance came from the gravitational wave detection, which told us it was produced by a nearby neutron star merger,” says Elena Pian, from the Instituto Nazionale di Astrofisica, Bologna.

Hours later, after the astronomy community had been alerted, telescopes over the world and in space were pointing towards the remaining blob. Studying light coming from the blob showed it contained rare elements such as gold and platinum, confirming theories of how these heavier elements are formed.

What was left after the burst is not completely known. It is probably a black hole, because neutron stars heavier than twice the mass of our Sun have never been seen. “It’s [probably] the first observation of a black hole being created where there was none before, which is pretty darn cool,” Shoemaker says.

The discovery has been heralded by many as the start of a new era of astrophysics, focussed on multi-messenger gravitational astrophysics. From now on it is hoped gravitational and electromagnetic observatories will continue to work together to look at the same source, recreating the incredible global effort put into this discovery.

It is hoped gravitational waves may be a way to explain a lot of mysteries. These include why the Universe is expanding at an accelerating rate, and the composition of dark energy, an elusive, mysterious substance that makes up roughly 70 per cent of the Universe.

“This is a whole new window into the Universe,” says Marcelle Soares-Santos, assistant professor of physics at Brandeis University. “This is beyond my wildest dreams.”